1.1. The abstract setting

Let us denote the abstract nuclear variables by $y= (y_1, \dots , y_m) \in {\mathbb{R}}^m$ , for some $m \in \mathbb{N}$ , fixed but arbitrary. Let us fix a Hilbert space ${\mathcal{H}}_{\mathrm {el}}$ for the electronic configuration. Consider the Hilbert space ${\mathcal{H}} = L^2({\mathbb{R}}^m, {\mathcal{H}}_{\mathrm {el}})$ for the whole (abstract) molecular dynamics. To separate the nuclear variable $y \in {\mathbb{R}}^m$ from the electronic variables in ${\mathcal{H}}_{\mathrm {el}}$ , we use the formalism of direct integrals of Hilbert spaces (following, for instance, [Reference Jecko28–Reference Hoever31]).

We consider the whole space ${\mathcal{H}} = L^2({\mathbb{R}}^m, {\mathcal{H}}_{\mathrm {el}})$ where the molecular dynamics take place as the fibred space of square integrable functions of $y \in {\mathbb{R}}^m$ with values in the Hilbert space ${\mathcal{H}}_{\mathrm {el}}$ . A vector $\Psi \in {\mathcal{H}}$ can be written as the vector direct integral

(1.8) \begin{equation} \Psi = \int _{{\mathbb{R}}^m}^\oplus \Psi (y) dy, \end{equation}

so that the usual inner product in $\mathcal{H}$ reads

(1.9) \begin{equation} \langle \Psi , \Phi \rangle = \int _{{\mathbb{R}}^m} \langle \Psi (y), \Phi (y) \rangle _{{\mathcal{H}}_{\mathrm {el}}} dy, \quad \Psi , \Phi \in {\mathcal{H}}. \end{equation}

With this notation, we can write $\mathcal{H}$ as a constant fibre direct integral or a direct integral of Hilbert spaces,

(1.10) \begin{equation} {\mathcal{H}} = \int _{{\mathbb{R}}^m}^\oplus {\mathcal{H}}_{\mathrm {el}} dy. \end{equation}

Given a family of operators $\left \{ A(y)\right \}_{y\in {\mathbb{R}}^m}$ on ${\mathcal{H}}_{\mathrm {el}}$ , we can define an operator $A$ acting on $\mathcal{H}$ by setting

(1.11) \begin{equation} (A\Psi )(y) = A(y)\Psi (y), \quad y\in {\mathbb{R}}^m, \quad \Psi \in {\mathcal{H}}, \end{equation}

whenever this definition makes sense. We call such operators fibred (over ${\mathbb{R}}^m$ ) and write them as

(1.12) \begin{equation} A = \int ^{\oplus }_{{\mathbb{R}}^m} A(y) dy, \end{equation}

with $A(y)$ called the fibre operators.

In the spirit of the BOA described above, let $H_\kappa$ be the operator family on the space $\mathcal{H}$ given by

(1.13) \begin{equation} H_\kappa = - \kappa ^2 \Delta _y + H_{\mathrm {bo}}, \quad \:\mathrm { with}\ H_{\mathrm {bo}} = \int _{{\mathbb{R}}^{m}}^\oplus H(y) dy, \end{equation}

where $H_{\mathrm {bo}}$ is an abstract Born–Oppenheimer operator, meaning that we assume $H(y)$ is a smooth family of self-adjoint operators with isolated and simple lowest eigenvalues $E(y)$ and associated eigenvectors $\psi _\circ (y)$ . For simplicity, we assume $H(y)$ is real.

Now, consider the Schrödinger equation associated with this abstract molecular Hamiltonian in the molecular space $\mathcal{H}$ , i.e.

(1.14) \begin{equation} \left \{ \begin{array}{l@{\quad}l} i \kappa \partial _t \Psi = H_{\kappa } \Psi , & \mathrm {in } {\mathcal{H}}, \\[3pt] \Psi \vert _{t=0} = \Psi _0 \in {\mathcal{H}}. & \end{array} \right. \end{equation}

We seek the effective nuclear dynamics in the nuclear variables $y\in {\mathbb{R}}^m$ only. To find an effective nuclear dynamics, we project out the electronic degrees of freedom in (1.14). This amounts to passing from the full Hilbert space $\mathcal{H}$ to the space of nuclear states ${\mathcal{H}}_{\text{nucl}} = L^2({\mathbb{R}}^m)$ in the following way. Writing $ t \mapsto \Psi (t) = \int ^{\oplus }_{{\mathbb{R}}^m} \Psi (y,t) dy$ in $\mathcal{H}$ , we define a corresponding path $t \mapsto f(t)$ in ${\mathcal{H}}_{\mathrm {nucl}}$ as

(1.15) \begin{equation} f(y, t) \;:\!=\; \langle \psi _\circ (y), \Psi (y, t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}. \end{equation}

The function $f(y, t)$ can be interpreted as the state of the nuclei at time $t$ , given that the electrons are in the ground state. We expect the function $f(y, t)$ , associated with $\Psi (t)$ , solution to the Schrödinger equation (1.14), to satisfy an effective equation of the form

(1.16) \begin{equation} \left \{ \begin{array}{l} i \kappa \partial _t f = h_{\mathrm {eff}}^\kappa\, f, \\[3pt] f |_{t=0} = f_0\in {\mathcal{H}}_{\mathrm {nucl}}, \end{array} \right. \end{equation}

with some effective nuclear Hamiltonian $h_{\mathrm {eff}}^\kappa$ to be determined in terms of the abstract molecular Hamiltonian (1.13). It will turn out that $h_{\mathrm {eff}}^\kappa$ is a non-local operator acting on functions $f \in C({\mathbb{R}}, {\mathcal{H}}_{\text{nucl}})$ . Once the effective nuclear Hamiltonian is found, we explain how to reconstruct the original molecular wave function $\Psi (t)$ from the reduced effective nuclear wave function $f(t)$ .

1.2. Effective nuclear dynamics in the abstract setting

Let $H_{\kappa }$ be given by (1.13), and let

(1.17) \begin{equation} U_t = e^{- i H_\kappa t / \kappa }, \end{equation}

be the associated propagator. Denote the kinetic energy of the nuclei as

(1.18) \begin{equation} T = - \kappa ^2 \Delta _y, \quad \mathrm {in }\; {\mathbb{R}}^m, \end{equation}

where $\Delta _y$ is the usual (rescaled) Laplacian in nuclear variables.

Our first result is the time-dependent Born–Oppenheimer correction to first order. Let $(\psi _\circ\, f)(y) = \psi _\circ (y) f(y)$ and $\langle t \rangle = (1 + \left |t\right |^2)^{1/2}$ . For $s \in \mathbb{N}$ , the spaces $H^{s, \kappa }_{\mathrm {nucl}}$ and ${\mathcal{H}}_{\mathrm {el}} H^{s, \kappa }_{\mathrm {nucl}}$ are defined in Section 1.3. For $s,s' \in \mathbb{N}$ , we shall say that the operator $F \;:\; {\mathcal{H}}_{\mathrm {el}} H^{s, \kappa }_{\mathrm {nucl}} \rightarrow {\mathcal{H}}_{\mathrm {el}} H^{s', \kappa }_{\mathrm {nucl}}$ is $O_{{\mathcal{L}}_{s,s'}}(a),$ for some $a \geq 0$ if there exists a constant $C\gt 0$ such that

(1.19) \begin{equation} \Vert F \phi \Vert _{{\mathcal{H}}_{\mathrm {el}} H^{s,\kappa }_{\mathrm {nucl}}} \leq C \Vert \phi \Vert _{{\mathcal{H}}_{\mathrm {el}}H^{s',\kappa }_{\mathrm {nucl}}}, \quad \forall \phi \in {\mathcal{H}}_{\mathrm {el}} H^{s,\kappa }_{\mathrm {nucl}}. \end{equation}

With the assumptions formulated in Section 1.3 and the previous notation, we have our first result.

Theorem 1.1. (Time-dependent BOA to order $\kappa$ .) Let Assumptions [A1]–[A4] below hold. Then, for any $f \in H^{2, \kappa }_{\mathrm {nucl}}$ , we have

(1.20) \begin{equation} U_t \psi _\circ\, f = \psi _\circ \left ( e^{-i h_{\mathrm {eff}} t /\kappa } f \right ) + O_{{\mathcal{L}}_{2,0}}(\kappa \langle t \rangle ^3), \end{equation}

where

(1.21) \begin{equation} h_{\mathrm {eff}} = T + E + \kappa ^2 v, \end{equation}

with $E$ and $v$ being multiplication operators by the functions $E(y)$ and

(1.22) \begin{equation} v(y)\;:\!=\; \lVert {\nabla _{y} \psi _\circ }\rVert ^2_{{\mathcal{H}}_{\mathrm {el}}}. \end{equation}

Theorem1.1 is proven in Section 3. It states that the solution of the Schrödinger IVP (1.14) with the initial condition $\Psi _0 = \psi _\circ\, f_0$ can be approximated, up to the order $O_{{\mathcal{L}}_{2,0}}(\kappa \langle t \rangle ^3)$ , by $\Psi (t) = \psi _\circ\, f(t)$ , where $f(t)$ is the solution of the effective equation

(1.23) \begin{equation} i \kappa \partial _t f = h_{\mathrm {eff}} f, \quad f |_{t=0} = f_0. \end{equation}

The last term in Equation (1.21) can be dropped. We keep it because it arises naturally and the corresponding derivation is used later on.

Our next result gives the effective nuclear dynamics (with effective nuclear Hamiltonian $h_{\mathrm {eff}}^\kappa$ ) to an arbitrary order beyond the leading order given in (1.21).

Let $P(y)$ denote the eigenprojection associated with the eigenvalue $E(y)$ , i.e. the orthogonal projection onto the ground state $\psi _\circ (y)$ of $H(y)$ ,

(1.24) \begin{equation} P(y)\phi = \psi _\circ (y) \langle \psi _\circ (y), \phi \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \quad \forall \phi \in {\mathcal{H}}_{\mathrm {el}}. \end{equation}

Using projections (1.24), we define the fibred operator

(1.25) \begin{equation} P = \int ^\oplus _{{\mathbb{R}}^m} P(y) dy, \quad P\;:\; {\mathcal{H}} \to \mathrm {Ran }\;P \equiv \psi _\circ \otimes {\mathcal{H}}_{\mathrm {nucl}} \cong {\mathcal{H}}_{\mathrm {nucl}}. \end{equation}

It is easy to see that $P$ is an orthogonal projection on ${\mathcal{H}} = L^2({\mathbb{R}}^m, {\mathcal{H}}_{\mathrm {el}})$ of infinite rank. Further, let $\overline {P}\;:\!=\;1-P$ , the orthogonal complement to $P$ . We define the reduced resolvent

(1.26) \begin{equation} {\overline {R}} = {\overline {P}} ((H_{\mathrm {bo}} - E)|_{{\mathrm {Ran}}\, {\overline {P}}})^{-1} {\overline {P}}, \end{equation}

which is well-defined thanks to Assumption [A4] (see Lemma 3.1). Let us introduce the operator

(1.27) \begin{equation} \overline {H} = {\overline {P}} H_{\kappa } {\overline {P}}, \end{equation}

which, by self-adjointness (see Appendix B), generates the propagator

(1.28) \begin{equation} \overline {U}_t = e^{- i \overline {H} t / \kappa }, \quad t \in {\mathbb{R}}. \end{equation}

Given an interval $I \subseteq \mathbb{R}$ , let us define the map $Q_P \;:\; C(I,{\mathcal{H}}_{\mathrm {nucl}}) \to C(I, {\mathcal{H}})$ as

(1.29) \begin{align} &(Q_P f)(t) \;:\!=\; \psi _\circ\, f(t) - \frac {i}{ \kappa } \int _0^t \overline {U}_{t-s} \overline {P} H_{\kappa } P\psi _\circ\, f(s)\, \text{d}s, \quad t \in I, \end{align}

where we denote $f(t)(y) = f(y,t)$ and $\Psi (t)(y) = \Psi (y,t) \in {\mathcal{H}}_{\mathrm {el}}$ . The main result of this paper is the following theorem.

Theorem 1.2. (Effective nuclear dynamics.) Let Assumptions [A1]–[A4] below hold. Let $H_{\kappa }$ be given by ( 1.13 ). Let $\Psi _0 = \psi _\circ\, f_0$ for some $f_0 \in {\mathcal{H}}_{\mathrm {nucl}}$ . We have the following:

(a) If $\Psi = \Psi (t)$ satisfies the Schrödinger equation ( 1.14 ) on ${\mathcal{H}} = L^2({\mathbb{R}}^m, {\mathcal{H}}_{\mathrm {el}})$ with $\Psi (0) = \Psi _0$ , then $f(t) \;:\!=\; \langle \psi _\circ , \Psi (t) \rangle _{\mathcal{H}_{el}}$ (see ( 1.15 )) obeys the Schrödinger equation ( 1.16 ),

(1.30) \begin{equation} i \kappa \partial _t f = h_{\mathrm {eff}}^\kappa\, f, \quad f(0) = f_0, \end{equation}

with the effective (non-local) nuclear Hamiltonian

(1.31) \begin{equation} h_{\mathrm {eff}}^\kappa = T + E + \kappa ^2 v + w^\kappa , \end{equation}

where $E$ and $v$ are multiplication operators by the functions $E(y)$ and ( 1.22 ), and $w^{\kappa }$ is the non-local operator acting on functions $f \in C({\mathbb{R}}, L^2({\mathbb{R}}^m))$ given by

(1.32) \begin{equation} w^\kappa [f](t) = - \frac {i}{ \kappa } \int _0^t \langle \psi _\circ , P T \overline {P} \:\overline {U}_{t-s} \overline {P} T P \psi _\circ\, f(s) \rangle _{{\mathcal{H}}_{\mathrm {el}}}\, \text{d}s. \end{equation}

Conversely, if $f(t)$ obeys ( 1.30 ) with the effective nuclear Hamiltonian $h_{\mathrm {eff}}^\kappa$ , then $Q_Pf(t)$ obeys ( 1.7 ) with initial conditions $Q_Pf(0) = \psi _\circ\, f_0$ .

(b) Let $n \geqslant 2$ be an integer. Assume that $f_0 \in H^{s+2n, \kappa }_{\mathrm {nucl}}$ , for some integer $0 \leqslant s \leqslant k_A - 2n - 2$ , and let $f$ be the corresponding solution of ( 1.30 ) in $B^{s+2n+2}_\tau \;:\!=\; L^\infty (0,\tau ;\; H^{s+2n+2, \kappa }_{\mathrm {nucl}})$ (see ( 1.69 )) for some $\tau \gt 0$ . Then, for every $0 \leq t \leq \tau$ , the electronic feedback operator $w^\kappa$ , defined in ( 1.32 ), acting on $f$ admits the following expansion:

(1.33) \begin{equation} w^\kappa [f](t) = \sum _{j=1}^{n-1} (\!-\!i \kappa )^{j+1} \left (w_jf(t) - \tilde w_j(t) f_0 \right ) + (\!-\!i\kappa )^{n+1} w^\kappa _n[f](t), \end{equation}

where for $j \geqslant 1, w_j$ is an operator satisfying the estimate

(1.34) \begin{equation} \lVert {w_j u}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim \lVert {u}\rVert _{H^{s+j+1, \kappa }_{\mathrm {nucl}}}, \quad \forall u \in H^{s+j+1, \kappa }_{\mathrm {nucl}}, \end{equation}

$\tilde w_j(t)$ is a local operator family on $L^2({\mathbb{R}}^m)$ satisfying the estimates

(1.35) \begin{equation} \lVert {\tilde w_j(t) u}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim e^{C \tau } \lVert {u}\rVert _{H^{s+2j, \kappa }_{\mathrm {nucl}}}, \quad \forall u \in H^{s+2j, \kappa }_{\mathrm {nucl}}, \quad 0 \leq t \leq \tau , \end{equation}

and $w^{\kappa }_n[f]$ is a non-local in $t$ operator satisfying the estimate

(1.36) \begin{equation} \lVert {w^{\kappa }_n [f](t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim e^{C \tau } \left ( \lVert {f}\rVert _{B^{s+2n+2}_\tau } + \lVert {f_0}\rVert _{H^{s+2n, \kappa }_{\mathrm {nucl}}} \right ), \quad 0 \leq t \leq \tau. \end{equation}

The operators $w_1$ , $\tilde w_1(t)$ and $w_2^\kappa$ are given explicitly in ( 4.29 )–( 4.31 ), respectively. The higher-order operators can be computed explicitly using the expansion scheme described in Section 4.3 .

In part (b) of Theorem1.2 above, we have assumed that if $f_0 \in H^{s+2n, \kappa }_{\mathrm {nucl}}$ , for some $s\gt 0$ and $n \geqslant 1$ , the corresponding solution $f$ of (1.30) exists and satisfies $ f \in {B^{s+2n+2}_\tau }$ for some $\tau \gt 0$ . While the existence of such $f$ is clear from the existence of the full dynamics $\Psi$ , the existence of the effective dynamics stemming from $f_0$ alone and satisfying $ f \in {B^{s+6}_T}$ is not completely straightforward. This can be addressed using the theory of existence of solutions of integro-differential Volterra type equations (see [Reference Desch and Schappacher32–Reference Oka36] and for a comprehensive review, [Reference Tismane, Bounit and Fadili37, Section 2]).

Now we address the question of effective nuclear dynamics at second order. In light of part (b) of Theorem1.2, this is now possible. Our next theorem says that the solution $\Psi (t)$ of the Schrödinger IVP (1.14) with the initial condition $\Psi _0 = \psi _\circ\, f_0$ is of the form

(1.37) \begin{equation} \Psi (t) = Q_P \tilde {f}(t) + O_{{\mathcal{L}}_{7,0}}(\kappa ^2 e^{Ct}), \end{equation}

for some constant $C \gt 0$ , where $\tilde {f}(t)$ is the solution of the effective equation given below.

Theorem 1.3. (Effective nuclear dynamics to order $\kappa ^2$ .) Let Assumptions [A1]–[A4] below hold with $H_{\kappa }$ given by ( 1.13 ). Let $\Psi (t)$ satisfy the Schrödinger equation ( 1.14 ) on ${\mathcal{H}} = L^2({\mathbb{R}}^m, {\mathcal{H}}_{\mathrm {el}})$ with $\Psi (0) = \psi _\circ\, f_0$ for some $f_0 \in H^{7, \kappa }_{\mathrm {nucl}}$ . Then, there exists a $\kappa _0 = \kappa _0 (\delta , m, \lVert {\nabla _{y_j} H_{\mathrm {bo}}}\rVert , \dots , \lVert {\nabla _{y_k} \psi _\circ }\rVert , \dots\!)$ such that for all $\kappa \in (0, \kappa _0)$ , there exists a constant $C \gt 0$ such that

(1.38) \begin{equation} \lVert {\Psi (t) - Q_P \tilde {f}(t)}\rVert \leqslant \kappa ^2 C e^{C \tau } \lVert {f_0}\rVert _{H^{7, \kappa }_{\mathrm {nucl}}}, \quad 0 \leqslant t \leq \tau , \end{equation}

where $\tilde {f}(t)$ is a solution of the autonomous equation

(1.39) \begin{equation} i \kappa \partial _t \tilde {f} = h_{\mathrm {eff}}^{(2)} \tilde {f}, \quad h_{\mathrm {eff}}^{(2)} \;:\!=\; T + E + \kappa ^2 v - \kappa ^2 w_1, \end{equation}

with the initial condition $\tilde {f} |_{t=0} = f_0$ , where $w_1$ (see ( 1.33 )) is given explicitly as

(1.40) \begin{equation} w_1 = -\frac {1}{\kappa ^2} \langle \psi _\circ , P T {\overline {P}}{\kern1pt} {\overline {R}}{\kern1pt} {\overline {P}} T P \psi _\circ \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \quad w_1 = O_{{\mathcal{L}}_{s+2,s}}(1). \end{equation}

As $T + E = O_{{\mathcal{L}}_{2,0}}(1)$ , the operator $T + E$ is the time-dependent BOA to first order. For $\kappa \in (0, \kappa _0)$ , the operator $h_{\mathrm {eff}}^{(2)}$ is self-adjoint on ${\mathcal{H}}_{\mathrm {nucl}}$ with domain $H^{2,\kappa }_{\mathrm {nucl}}$ .

Theorems1.2 and 1.3 state that $\kappa ^2 v - \kappa ^2 w_1$ is the correct second-order time-dependent Born–Oppenheimer correction. By iterating the expansion (1.33) to higher orders in $\kappa$ (see (4.51)), higher-order corrections to the time-dependent BOA can be derived, and Theorem1.3 can be extended to these higher orders.

(1.39) was first obtained in [Reference Panati, Spohn and Teufel13, Reference Panati, Spohn and Teufel21, Reference Weigert and Littlejohn38]. In the physics literature, a similar equation was first obtained by Weigert and Littlejohn in [Reference Gustafson and Sigal39] for matrix-valued Hamiltonians.

1.3. Assumptions in the abstract setting

In order to track the dependence with respect to $\kappa$ , we introduce the rescaled differential operators (in nuclear variables)

(1.41) \begin{equation} D_{y} \;:\!=\; -i\kappa \partial _{y}, \quad D^\alpha = \prod _{j=1}^m D^{\alpha _j}_{y_j}, \end{equation}

and we denote $\partial _y^\alpha = \partial ^{\alpha _1}_{y_1} \dots \partial ^{\alpha _m}_{y_m}$ for any $\alpha = (\alpha _1, \dots , \alpha _m) \in \mathbb{N}^m$ with $\left |\alpha \right | = \sum _{j=1}^m \alpha _j$ . Writing ${\mathcal{H}}_{\mathrm {nucl}} \equiv L^2({\mathbb{R}}^m)$ , we introduce the $\kappa$ -scaled Sobolev spaces:

(1.42) \begin{align} H^{s, \kappa }_{\mathrm {nucl}} &= \bigg\{\phi \in L^2({\mathbb{R}}^m) \::\: \lVert {\phi }\rVert _{H^{s, \kappa }_{\mathrm {nucl}}}^2 = \sum _{\left |\alpha \right | = s} \lVert {D_{y}^\alpha \phi }\rVert _{{\mathcal{H}}_{\mathrm {nucl}}}^2 + \lVert {\phi }\rVert _{{\mathcal{H}}_{\mathrm {nucl}}}^2 \lt \infty \bigg\}. \end{align}

with the usual convention $H^{0, \kappa }_{\mathrm {nucl}} = {\mathcal{H}}_{\mathrm {nucl}} = L^2({\mathbb{R}}^m)$ .

Assumption [A1]. The operators $H(y)$ are self-adjoint on ${\mathcal{H}}_{\mathrm {el}}$ with a dense domain ${\mathcal{D}}_\circ$ independent of $y$ , and there exist constants $\gamma$ and $k_A \geqslant 7$ , independent of $y$ , such that

(1.43) \begin{equation} \langle \phi , (H(y) + \gamma ) \phi \rangle _{{\mathcal{H}}_{\mathrm {el}}} \geq \gamma \Vert \phi \Vert ^2_{{\mathcal{H}}_{\mathrm {el}}}, \quad \phi \in {\mathcal{H}}_{\mathrm {el}}, \end{equation}

and

(1.44) \begin{equation} \lVert {\partial _y^{\alpha } H(y)}\rVert _{{\mathcal{L}}({\mathcal{H}}_{\mathrm {el}})} \leqslant C(\alpha ), \quad \forall \alpha \text{ with } 1 \leqslant \left |\alpha \right | \leqslant k_A \text{ and } y \in {\mathbb{R}}^m, \end{equation}

where the constant $C(\alpha )$ depends only on $\alpha$ .

Assumption [A2]. For all $\kappa \gt 0$ , the operators $H_{\kappa }$ are self-adjoint on $\mathcal{H}$ with a dense domain ${\mathcal{D}}(H_\kappa )$ independent of $\kappa$ ,

(1.45) \begin{equation} {\mathcal{D}}(H_{\kappa }) = {\mathcal{H}}_{\mathrm {el}} \otimes H^{2,\kappa }_{\mathrm {nucl}} \cap {\mathcal{D}}_\circ \otimes {\mathcal{H}}_{\mathrm {nucl}}. \end{equation}

Here, ${\mathcal{D}}_\circ$ is the $y$ -independent domain of the fibres $H(y)$ from Assumption [A1].

Assumption [A3]. The operators $H(y)$ have unique ground states, which we denote $\psi _\circ (y)$ . The corresponding non-degenerate ground state energy, $E(y)$ , satisfies

(1.46) \begin{equation} H(y) \psi _\circ (y) = E(y) \psi _\circ (y), \quad \lVert {\psi _\circ (y)}\rVert _{{\mathcal{H}}_{\mathrm {el}}} = 1, \quad \forall y \in {\mathbb{R}}^m. \end{equation}

Since $H(y)$ is real, we can assume $\psi _\circ (y)$ to be real.

Assumption [A4]. There is a gap between the ground state energy $E(y)$ of $H(y)$ and the rest of the spectrum of $H(y)$ , which is positive uniformly for $y \in {\mathbb{R}}^m$ , i.e. there exists a $\delta \gt 0$ such that

(1.47) \begin{equation} \inf _{y \in {\mathbb{R}}^m} \left \{ \vert \xi - E(y) \vert ; \, \xi \in \sigma (H(y)) \setminus \{E(y)\} \right \} \geqslant \delta \gt 0. \end{equation}

1.4. Main results in the molecular case

Now, we specify the above construction to a molecular system as described by (1.1). We view the space $L^2_{\mathrm {mol}}$ as the space $L^2({\mathbb{R}}^{3M}, L^2_{\mathrm {el}})$ , with $L^2_{\mathrm {el}}$ the subspace of $L^2({\mathbb{R}}^{3N})$ carrying an irreducible representation of the permutation symmetry group $S_N$ of $N$ indices, as already mentioned above. We rewrite (1.1) as

(1.48) \begin{equation} H_{\text{mol}} = T + H_{\mathrm {bo}}, \quad \mathrm { with } \quad H_{\mathrm {bo}} = \int _{\mathbb{R}^{3M}}^\oplus H(y) dy, \end{equation}

where the operators $H(y)$ and $T$ acting on $L^2_{\mathrm {el}}$ and $L^2_{\mathrm {nucl}}$ , respectively, are defined as

(1.49) \begin{align} H(y) &= - \sum _{j=1}^N \frac {1}{2} \Delta _{x_j} + V_e(x) + V_{en}(x,y) + V_n(y), \end{align}

(1.50) \begin{align} T &= - \sum _{j=1}^M \frac {1}{2 m_j} \Delta _{y_j} = -\kappa ^2 \sum _{j=1}^M \frac {1}{2 m_j'} \Delta _{y_j}, \quad m_j' = \kappa ^2 m_j = \frac {m_j}{\min _{k} m_k}.\\[8pt] \nonumber \end{align}

The operator $H(y)$ on $L^2_{\mathrm {el}}$ is supposed to satisfy Assumptions [A1]–[A4]. Note that, by introducing the inner product

(1.51) \begin{equation} (y, \tilde {y}) = 2 \sum _{j=1}^M m_j' y_j \cdot \tilde {y}_j \end{equation}

in ${\mathbb{R}}^{3M}$ , the operator $T$ can be written as

(1.52) \begin{equation} T = - \kappa ^2 \Delta _y, \end{equation}

where $\Delta _y$ is the Laplace-Beltrami operator in the metric (1.51).

The electrons have charges $-e$ , and the nuclei have charges $Z_j e$ , where $j = 1, \dots , M$ . The electrons are modelled as point charges, and the electronic repulsion is given by the Coulomb potential energy

(1.53) \begin{equation} V_{\mathrm {e}}(x) = \sum _{i=1}^{N-1} \sum _{j=i+1}^N \frac {e^2}{\lvert {x_i - x_j}\rvert }. \end{equation}

For technical and physical reasons, we shall model the nuclei as smeared rigid charge distributions $\rho \in C^\infty _c(\mathbb{R}^3)$ , $\rho \geqslant 0$ , and $\lVert {\rho }\rVert _{L^1} = 1$ , as opposed to the point charges. These are referred to in the literature as form-factors (see [Reference Simon40] and [Reference Weigert and Littlejohn38]). The potential for nuclear repulsion becomes

(1.54) \begin{equation} V_{\mathrm {n}}(y) = \sum _{i=1}^{M-1} \sum _{j=i+1}^M \int _{\mathbb{R}^6} \frac {e^2 Z_i Z_j \rho (z - y_i) \rho (z' - y_j)}{\lvert {z - z'}\rvert } dz dz', \end{equation}

and the attractive potential between electrons and nuclei becomes

(1.55) \begin{equation} V_{\mathrm {en}}(x,y) = - \sum _{i=1}^M \sum _{j=1}^N \int _{\mathbb{R}^3} \frac {e^2 Z_i\rho (z - y_i)}{\lvert {z - x_j}\rvert } dz. \end{equation}

Observe that $V_{\mathrm {n}}$ and $V_{\mathrm {en}}$ are bounded and that $x \mapsto V_{\mathrm {en}}(x,\cdot )$ is smooth, and $V_{\mathrm {en}}(x,\cdot ) \in L^\infty ({\mathbb{R}}^{3M})$ for all $x \in {\mathbb{R}}^{3N}$ .

We apply Theorems1.1–1.3 to the molecular system given by (1.7) with (1.48)–(1.55), identifying $m = 3M$ , ${\mathcal{H}} = L^2_{\mathrm {mol}}$ , ${\mathcal{H}}_{\mathrm {nucl}} = L^2_{\mathrm {nucl}}$ and ${\mathcal{H}}_{\mathrm {el}} = L^2_{\mathrm {el}}$ to obtain the following theorems.

To derive Theorems1.4–1.6 from Theorems1.1–1.3, we have to show that the family of operators in (1.49) satisfies Assumptions [A1] and [A2].

The self-adjointness of $H_{\text{mol}}$ and $H_{\mathrm {bo}}$ is standard (see, for example, [Reference Simon40–Reference Avron, Seiler and Yaffe43]). The differentiability follows from the fact that

(1.56) \begin{equation} \partial _y^\alpha H(y) = \partial _y^\alpha \left (V_n(y) + V_{en}(x,y) \right ) \end{equation}

where $V_n(y)$ and $V_{en}(x,y)$ , given by (1.54) and (1.55), are bounded and self-adjoint multiplication operators on ${\mathcal{H}}_{\mathrm {el}}$ by smooth and real functions of $y$ . The semi-boundedness is also well-known (see [Reference Hunziker and Sigal41, Theorem 7.1.15] and the references above).

1.5. Earlier results and remarks

The second order correction to the BO propagator was first derived rigorously in [Reference Panati, Spohn and Teufel13] (see also [Reference Teufel12, Reference Spohn and Teufel14, Reference Panati, Spohn and Teufel21]).

A different integration by parts method was introduced in [Reference Brummelhuis and Nourrigat44] and applied to the time-dependent BO by Panati, Spohn, and Teufel [Reference Panati, Spohn and Teufel13, Reference Spohn and Teufel14, Reference Panati, Spohn and Teufel21, Reference Spohn and Teufel22, Reference Weigert and Littlejohn38] to derive first-order corrections. For higher-order corrections, Panati, Spohn, and Teufel made use of the pseudo-differential calculus with operator-valued symbols (see also [Reference Martinez and Sordoni19, Reference Emmrich and Weinstein45–Reference Lasser and Lubich48]).

The novel contributions of this paper compared to earlier work are

• • Our approximation of the molecular propagator by an adiabatic one holds for a large class of initial conditions.

• • We develop a non-abelian integration by parts (NAIP) method, which iterates to higher orders, circumventing the need for complex pseudo-differential calculus used in prior approaches.

In Theorem1.3, we derive the effective second-order corrections, matching those derived by Teufel et al. (see, for example, [Reference Panati, Spohn and Teufel21, Section 2.2]).

Hagedorn et al. follow a distinct approach, constructing semiclassical wavepackets approximating the exact solutions [Reference Hagedorn and Joye18, Reference Hagedorn23–Reference Hagedorn and Joye27]. For computational semiclassical approaches, see [Reference Jin, Markowich and Sparber49, Reference Gherghe50].

1.6. Organisation of the paper

This paper is organised as follows. In Section 2 we develop the non-abelian integration by parts, NAIP, which is our central technique used to obtain Theorems1.1–1.3. Then, in Section 3 we prove Theorem1.1, relying upon technical results that are proven in Appendices A–D.

In Section 4, we prove Theorem1.2. Section 4.1 deals with part (a), in Sections 4.2–4.4, we focus on the proof of part (b), which also requires the NAIP developed in Section 2.

In Section 5, we prove Theorem1.3. Again, we will make use of the NAIP of Section 2.

In Appendix A, we collect some basic results on fibred operators, while in Appendix B, we establish the self-adjointness of $P H_\kappa P$ and $\overline {H} = {\overline {P}} H_\kappa {\overline {P}}$ . Appendices C and D deal, respectively, with commutator estimates and propagators.

1.7. Notation

In this section, we gather some of the notation used throughout the paper. We reserve the symbols $f, g$ for elements of $C({\mathbb{R}}, L^2({\mathbb{R}}^m))$ and $u, v$ for functions of $L^2({\mathbb{R}}^m)$ only.

We denote by $\mathcal{L}(\mathcal{H})$ and ${\mathcal{L}}({\mathcal{H}}_1, {\mathcal{H}}_2)$ the spaces of bounded linear operators on a Hilbert space $\mathcal{H}$ (equipped with the standard operator norm $\lVert {\cdot }\rVert$ ) and of bounded linear operators from ${\mathcal{H}}_1$ to ${\mathcal{H}}_2$ (equipped with the operator norm).

In the rest of this document, we will write ${\mathcal{H}}_1 {\mathcal{H}}_2$ to denote the tensor product ${\mathcal{H}}_1 \otimes {\mathcal{H}}_2$ between two Hilbert spaces ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ , equipped with the tensor product norm. We write ${\mathcal{H}}_{\mathrm {nucl}} = L^2({\mathbb{R}}^m)$ and ${\mathcal{H}} = L^2({\mathbb{R}}^m, {\mathcal{H}}_{\mathrm {el}}) = {\mathcal{H}}_{\mathrm {el}} {\mathcal{H}}_{\mathrm {nucl}}$ and use the Sobolev spaces (1.42) defined above and denote the tensor spaces

(1.68) \begin{align} {\mathcal{H}}_{\mathrm {el}} H^{s, \kappa }_{\mathrm {nucl}} &= {\mathcal{H}}_{\mathrm {el}} \otimes H^{s, \kappa }_{\mathrm {nucl}}({\mathbb{R}}^{m}) = H^{s, \kappa }_{\mathrm {nucl}} \left ({\mathbb{R}}^{m}, {\mathcal{H}}_{\mathrm {el}} \right ), \end{align}

with ${\mathcal{L}}_{r,s} = {\mathcal{L}}({\mathcal{H}}_{\mathrm {el}} H^{r,\kappa }_{\mathrm {nucl}}, {\mathcal{H}}_{\mathrm {el}} H^{s, \kappa }_{\mathrm {nucl}}).$

We shall make use of the time-dependent spaces $B^s_T = L^\infty (0, T;\; H^{s, \kappa }_{\mathrm {nucl}})$ , which are the Banach spaces equipped with norm

(1.69) \begin{equation} \lVert {g}\rVert _{B^s_T} = \sup _{t \in [0,T]} \lVert {g(t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}}. \end{equation}

The commutator of two unbounded operators on $\mathcal{H}$ , denoted $[\cdot ,\cdot ]$ and widely used throughout the article, is to be understood in the sense explained in Section C without explicit mention.

Finally, $A \lesssim B$ stands for the inequality $A \le C B$ , with some constant $C\gt 0$ independent of $\kappa$ . We also adopt the big $O$ notation in the sense that $A = O(\alpha )$ if there exists a constant $C \gt 0$ such that $\lVert {A}\rVert \leqslant C \alpha$ .

3.1. Onset of the proof

We now describe the approach and main steps of the proof of Theorem1.1 and outline the proof.

Given a solution $f$ to the effective dynamics generated by $h_{\mathrm {eff}}$ , we want to compare the evolution of $\psi _\circ\, f$ induced by the propagator $U_t$ , acting in all variables, with the evolution of $f$ restricted to the nuclear variable. This amounts at considering the difference

(3.1) \begin{equation} D(t) \;:\!=\; (U_t - U_t^P )P, \quad t \geq 0, \end{equation}

where $U_t = e^{-i H_\kappa t/\kappa}$ , see Equation (1.17), $U^P_t = e^{- i H^P t/\kappa}$ and, recall, $P$ stands for the (fibred) projections

(3.2) \begin{equation} P = \int ^{\oplus }_{{\mathbb{R}}^m} P(y) dy, \text{ with } P(y) = |\psi _\circ (y) \rangle \langle \psi _\circ (y) |, \end{equation}

or $P \Psi = \int ^\oplus _{{\mathbb{R}}^m} \psi _\circ (y) \langle \psi _\circ (y), \Psi (y) \rangle _{{\mathcal{H}}_{\mathrm {el}}} dy.$ Now, if $f = P\Psi$ , we want to estimate the difference $D(t)f$ acting on solutions to the effective Hamiltonian and show that this difference is of order $\kappa$ in the adequate spaces.

Thanks to Lemmas 3.2 and 3.3, we show that the difference $D(t)f$ can be recast as the integral in time of some operator-valued quantities. The next step is to prove that such integral terms are small using the NAIP of Lemma 2.2, which relies crucially on identity (2.1). We shall apply (2.1) to the self-adjoint operators $A = \frac {i}{\kappa } \overline {H}$ , $B = \frac {i}{\kappa } \overline {K}$ and the corresponding propagatorsFootnote ²

(3.3) \begin{equation} e^{As} = e^{i\overline {H} s/\kappa } \;=\!:\; \overline {U}_{-s}, \quad \overline {H} = {\overline {P}} H_\kappa {\overline {P}}, \end{equation}

and

(3.4) \begin{equation} e^{Bs} = e^{i \overline {K} s/\kappa } \;=\!:\; \overline {V}_{-s}, \quad \overline {K} = {\overline {P}} K {\overline {P}}, \quad K = E + T. \end{equation}

This choice, relying strongly on the physical grounds suggested by the BO approximation, is possible and effective since $\overline {H} - \overline {K} = \overline {H}_{\mathrm {bo}} - \overline {E}$ , with $\overline {H}_{\mathrm {bo}} \;:\!=\; {\overline {P}} H_{\mathrm {bo}} {\overline {P}}$ and $\overline {E} \;:\!=\; {\overline {P}} E {\overline {P}}$ , is invertible and therefore

(3.5) \begin{equation} (A-B)^{-1} = -i\kappa {\overline {P}} (\overline {H}_{\mathrm {bo}} - \overline {E})^{-1}{\overline {P}} \end{equation}

is well-defined thanks to the following lemma.

Lemma 3.1. Let Assumptions [A1]–[A4] hold. Then, the operator $\overline {H}_{\mathrm {bo}} - \overline {E} \;:\; \overline {\mathcal{D}} \to {\mathrm {Ran}}\, {{\overline {P}}}$ is invertible and its inverse, $\overline {R}$ (the reduced resolvent, see ( 1.26 )), is bounded uniformly in $y \in {\mathbb{R}}^m$ and satisfies

(3.6) \begin{equation} \lVert {\overline {R}}\rVert _{{\mathcal{L}}({\mathrm {Ran}}\, {{\overline {P}}})} \leqslant \frac {1}{\delta }, \end{equation}

where $\delta$ is given in ( 1.47 ).

Proof of Lemma 3.1. Since $E$ is a multiplication operator by the bounded real function $E(y)$ , $\overline {H}_{\mathrm {bo}} - \overline {E}$ is a self-adjoint operator with domain $\overline {\mathcal{D}}$ . Then by (1.13) and (1.47), we have

(3.7) \begin{equation} \overline {H}_{\mathrm {bo}} = \int ^\oplus _{{\mathbb{R}}^m} \overline {H}(y) dy \geqslant \int ^\oplus _{{\mathbb{R}}^m} E_1(y) \overline {P}(y) dy, \end{equation}

where $E_1(y) \;:\!=\; \inf \left (\sigma (H(y)) \setminus \{E(y)\} \right )$ . Let $\Psi \;:\!=\; \int ^\oplus _{{\mathbb{R}}^m} \Psi (y) dy$ be any element of $\mathrm {Ran }\;\overline {P}$ . Then, by the gap condition (1.47),

(3.8) \begin{align} \langle \Psi , (\overline {H}_{\mathrm {bo}}-\overline {E}) \Psi \rangle &\geqslant \int ^\oplus _{{\mathbb{R}}^m} \langle \Psi (y), (\overline {H}(y) - \overline {E}(y)) \Psi (y) \rangle _{{\mathcal{H}}_{\mathrm {el}}} dy \nonumber \\ &\geqslant \int ^\oplus _{{\mathbb{R}}^m} \langle \Psi (y), (E_1(y)-E(y)) \Psi (y) \rangle _{{\mathcal{H}}_{\mathrm {el}}} dy \nonumber \\ &\geqslant \int ^\oplus _{{\mathbb{R}}^m} \delta \lVert {\Psi (y)}\rVert ^2_{{\mathcal{H}}_{\mathrm {el}}} dy. \end{align}

Since $\int ^\oplus _{{\mathbb{R}}^m} \lVert {\Psi (y)}\rVert ^2_{{\mathcal{H}}_{\mathrm {el}}} dy = \lVert {\Psi }\rVert ^2$ , this gives

(3.9) \begin{equation} \langle \Psi , (\overline {H}_{\mathrm {bo}} - \overline {E}) \Psi \rangle \geqslant \delta \lVert {\Psi }\rVert ^2. \end{equation}

Since $\overline {H}_{\mathrm {bo}} - \overline {E}$ is self-adjoint, (3.9) yields $\sigma (\overline {H}_{\mathrm {bo}} - \overline {E}) \subset [\delta , \infty )$ and therefore the operator $\overline {H}_{\mathrm {bo}} - \overline {E}$ is invertible. Furthermore, we have by a standard spectral estimate,

(3.10) \begin{equation} \lVert {(\overline {H}_{\mathrm {bo}} - \overline {E})^{-1}}\rVert \leqslant \frac {1}{\mathrm {dist}(0, \sigma (\overline {H}_{\mathrm {bo}} - \overline {E}))}, \end{equation}

which implies the first estimate in (3.6).

In what follows, we present the subsequent steps of the proof as a sequence of lemmas, postponing the proof of some of the technical results to appendices.

3.2. Representation formulas

Our first step is the following lemma, which allows us to relate the self-adjoint operatorFootnote ³ $H^P$ to the effective Hamiltonian $h_{\mathrm {eff}}$ and also rephrase $d$ in terms of $\left (U_t - U^P_t \right ) P$ .

Lemma 3.2. Let Assumptions [A1]–[A4] hold.

a) For all $g \in H^{2,\kappa }_{\mathrm {nucl}}$ , the following identity holds:

(3.11) \begin{equation} P K P \psi _\circ g = \psi _\circ (K + \kappa ^2 v) g, \end{equation}

where $K = E + T$ and $v$ is the operator given by ( 1.22 ) when $\psi _\circ$ is real Footnote ⁴ . In particular, if $t \mapsto \Psi (t) \in {\mathcal{H}}$ and $f$ is given by ( 1.15 ), the operator $H^P$ satisfies

(3.12) \begin{equation} H^P \Psi = \psi _\circ h_{\mathrm {eff}} f, \end{equation}

where $h_{\mathrm {eff}} = T + E + \kappa ^2 v$ .

b) For all $\Psi \in {\mathrm {Ran}}\, P$ ,

(3.13) \begin{equation} U^P_t \Psi = \psi _\circ e^{- i h_{\mathrm {eff}} t/\kappa } f, \end{equation}

where $f = \langle \psi _\circ , \Psi \rangle _{{\mathcal{H}}_{\mathrm {el}}}$ and $h_{\mathrm {eff}} = T + E + \kappa ^2 v$ .

Proof of Lemma 3.2. Proof of a). We begin by observing that, if $g \in H^{2,\kappa }_{\mathrm {nucl}}$ , then

\begin{equation*} P(\psi _\circ g) = \psi _\circ g. \end{equation*}

Further, as $[K, \psi _\circ ] = [T, \psi _\circ ]$ , we may write

(3.14) \begin{align} K P \psi _\circ g = K \psi _\circ g = \psi _\circ K g + [T, \psi _\circ ] g. \end{align}

Hence, taking the projection $P$ on the left and using the definition of $P$ yields

(3.15) \begin{align} P K P \psi _\circ g &= P \psi _\circ K g + P [T, \psi _\circ ] g \nonumber \\ &= \psi _\circ K g + \psi _\circ \langle \psi _\circ , [T, \psi _\circ ] \rangle _{{\mathcal{H}}_{\mathrm {el}}} g. \end{align}

Now, we need to compute the last term in the identity above. Expanding the commutator we find

(3.16) \begin{align} [T, \psi _\circ ] g = - \kappa ^2 (\Delta _{y} \psi _\circ ) g - 2i \kappa (\nabla _{y} \psi _\circ ) D_{y} g. \end{align}

Henceforth, using that $ g \in H^{2,\kappa }_{\mathrm {nucl}}$ and is independent of the electronic variables,

(3.17) \begin{align} \langle \psi _\circ , [T, \psi _\circ ] \rangle _{{\mathcal{H}}_{\mathrm {el}}} g & = - \kappa ^2 \langle \psi _\circ , \Delta _{y} \psi _\circ \rangle _{{\mathcal{H}}_{\mathrm {el}}} g - 2 i \kappa \langle \psi _\circ , \nabla _{y} \psi _{y} \rangle _{{\mathcal{H}}_{\mathrm {el}}} D_{y} g \nonumber \\ &= - \kappa ^2 \left (\nabla _{y} \langle \psi _\circ , \nabla _{y} \psi _\circ \rangle _{{\mathcal{H}}_{\mathrm {el}}} \right ) g + \kappa ^2 \langle \nabla _{y} \psi _\circ , \nabla _{y} \psi _\circ \rangle _{{\mathcal{H}}_{\mathrm {el}}} g \nonumber \\ &\quad - 2 i \kappa \langle \psi _\circ , \nabla _{y} \psi _{\circ } \rangle _{{\mathcal{H}}_{\mathrm {el}}} D_{y} g. \end{align}

Since $\psi _\circ (y)$ is real and $\lVert {\psi _\circ }\rVert ^2_{{\mathcal{H}}_{\mathrm {el}}} = 1$ for any $y \in {\mathbb{R}}^{m}$ , we have $\langle \psi _\circ , \nabla _{y} \psi _{\circ } \rangle _{{\mathcal{H}}_{\mathrm {el}}} = 0$ (see (1.67)). Hence,

(3.18) \begin{equation} \langle \psi _\circ , [T, \psi _\circ ] \rangle _{{\mathcal{H}}_{\mathrm {el}}} g = \kappa ^2 \lVert {\nabla _{y} \psi _\circ }\rVert ^2_{{\mathcal{H}}_{\mathrm {el}}}g, \end{equation}

and we deduce

(3.19) \begin{equation} P K P \psi _\circ g = \psi _\circ K g + \kappa ^2 \lVert {\nabla _{y} \psi _\circ }\rVert ^2_{{\mathcal{H}}_{\mathrm {el}}} g, \end{equation}

which implies (3.11) with $v$ of the form (1.22).

Finally, (3.12) follows from (3.11). Indeed, given $t \mapsto \Psi (t) \in {\mathcal{H}}$ and $f$ defined by (1.15), we have that $P \Psi = \psi _\circ\, f$ . Then, using (3.11) with the choice $g\;:\!=\; f$ , we get

(3.20) \begin{align} H^P \Psi = P H P \Psi & = P (K - E + H_{\mathrm {bo}}) \psi _\circ\, f \nonumber \\ & = \psi _\circ (K + \kappa ^2 v) f + P (\!- E + H_{\mathrm {bo}}) \psi _\circ\, f. \end{align}

But the last term vanishes as

(3.21) \begin{align} P H_{\mathrm {bo}} \psi _\circ\, f = P H_{\mathrm {bo}} \psi _\circ\, f = P (H_{\mathrm {bo}} \psi _\circ ) f = P E \psi _\circ\, f, \end{align}

whence $P (\!- E + H_{\mathrm {bo}}) \psi _\circ\, f = 0$ and (3.12) follows.

Proof of b). The operator $h_{\mathrm {eff}}$ is self-adjoint on ${\mathcal{H}}_{\text{nucl}} = L^2({\mathbb{R}}^m)$ with domain $H^{2, \kappa }_{\mathrm {nucl}}$ by the standard Kato-Rellich theory, since $T$ is self-adjoint on the same domain and $E +\kappa ^2 v$ are bounded perturbations. Hence, the semigroup $e^{- i h_{\mathrm {eff}} t/\kappa }$ exists.

All $\Psi \in {\mathrm {Ran}}\, P$ can be written in the form $\Psi = \psi _\circ\, f_0$ , for some $f_0 \in {\mathcal{H}}_{\text{nucl}}$ . Denote $\Psi (t) = U^P_t \psi _0 f_0$ and using part a),

(3.22) \begin{equation} i \kappa \partial _t \Psi (t) = H^P \Psi (t) = \psi _\circ h_{\mathrm {eff}} f(t), \end{equation}

where $f(t) \;:\!=\; \langle \psi _\circ , \Psi (t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}$ . We have $f(0) = f_0$ and

(3.23) \begin{equation} i \kappa \partial _t f(t) = \langle \psi _\circ , H^P \Psi (t) \rangle = h_{\mathrm {eff}} f(t), \end{equation}

so the relation (3.13) follows.

We write the difference $\left (U_t - U^P_t \right ) P$ as an operator-valued integral in time in order to integrate by parts eventually.

Lemma 3.3. Let Assumptions [A1]–[A4] hold. Let $X = \frac {i}{\kappa } {\overline {P}} T P$ .

We have

(3.24) \begin{equation} \left (U_t - U^P_t \right ) P = U_t X_t, \end{equation}

where

(3.25) \begin{align} X_t &= i \kappa U_{-s} {\overline {R}} X U^P_s \bigg |_{0}^t - i \kappa \int _0^t U_{-s} X^* {\overline {R}} X P U^P_s\, \text{d}s \nonumber \\ &\quad - i \kappa \int _0^t U_{-s} X_2 U^P_s\, \text{d}s, \end{align}

where

(3.26) \begin{equation} X_2 = \frac {i}{\kappa } S X + \frac {i}{\kappa } {\overline {R}} \left ( \overline {K} X - X K^P \right ), \quad S = {\overline {R}} [H_{\mathrm {bo}} - E , T] {\overline {R}}. \end{equation}

Here, $K^P = P K P$ .

Proof of Lemma 3.3. We write the difference $\left (U_t - U^P_t \right ) P$ as the integral of the derivative (c.f. a Duhamel formula) to obtain

(3.27) \begin{align} \left (U_t - U^P_t \right ) P &= - U_t \left (U_{-t} U^P_t - \textbf {1} \right ) P \nonumber \\ &= - U_t \int _0^t \frac {d}{ds} \left (U_{-s} U^P_s \right ) P\, \text{d}s \nonumber \\ &= - \frac {i}{\kappa } \int _0^t U_{t-s} \left ( H - PHP \right ) P U^P_s\, \text{d}s, \end{align}

where we used that $U^P_s P = P U^P_s$ . Observe that $H_\kappa = H_{\mathrm {bo}} + T$ and $[H_{\mathrm {bo}}, P] = 0$ , so we have

(3.28) \begin{equation} \left (H_\kappa - H^P \right ) P = {\overline {P}} H_\kappa P = {\overline {P}} T P. \end{equation}

We use this on the right-hand side of (3.28) and use the relation $U_{t-s} = U_t U_{-s}$ to write (3.24) with

(3.29) \begin{equation} X_t = - \int _0^t U_{-s} X U^P_s\, \text{d}s, \quad X \;:\!=\; \frac {i}{\kappa } {\overline {P}} T P. \end{equation}

In order to place $X_t$ in a form where we can integrate by parts, we use the Duhamel formula again to write

(3.30) \begin{equation} U_{-s} {\overline {P}} = \overline {U}_{-s} {\overline {P}} + \frac {i}{\kappa } \int _0^s U_{-s+r} P H {\overline {P}}\,\overline {U}_{-r}\, \text{d}r. \end{equation}

Making the change of variables $r \mapsto a = s - r$ inside the integral above and using $- \frac {i}{\kappa } P H {\overline {P}} = - \frac {i}{\kappa } P T {\overline {P}} = X^*$ , we write

(3.31) \begin{equation} U_{-s} {\overline {P}} = Y_s \overline {U}_{-s} {\overline {P}}, \quad Y_s = \textbf {1} + \int _0^s U_{-a} X^* \overline {U}_{a} \text{d}a. \end{equation}

Introducing (3.32) into $X_t$ , we obtain

(3.32) \begin{equation} X_t \;:\!=\; - \int _0^t Y_s \overline {U}_{-s} X U^P_s\, \text{d}s, \quad \text{ with } \quad Y_s \;:\!=\; \textbf {1} + \int _0^s U_{-a} X^* \overline {U}_{a} \text{d}a. \end{equation}

We apply the NAIP formula Lemma 2.2 to $X_t$ in (3.25) with $G_s = Y_s$ , $A = \frac {i}{\kappa } \overline {H}$ , $B = \frac {i}{\kappa } \overline {K}$ (where recall $K = E + T$ and $\overline {K} = {\overline {P}} K {\overline {P}}$ ) and $F_s = X U^P_s$ . Hence,

(3.33) \begin{align} X_t &= i \kappa Y_s \overline {U}_{-s} {\overline {R}} X U^P_s \bigg |_{0}^t - \int _0^t Y_s \overline {U}_{-s} S X P U^P_s\, \text{d}s \nonumber \\ &\hspace {-30pt} - i \kappa \int _0^t \left [ \left (\partial _s Y_s\right ) \overline {U}_{-s} {\overline {R}} X U^P_s + Y_s \overline {U}_{-s} {\overline {R}} \left ( \frac {i}{\kappa } \overline {K} X U_s^P + \partial _s (X U_s^P) \right ) \right ]\, \text{d}s. \end{align}

We re-write the terms on the right-hand side into a more suitable form. First, we compute using that $H^P = K^P \equiv P K P$ ,

(3.34) \begin{align} \frac {i}{\kappa } \overline {K} X U_s^P + \partial _s (X U_s^P) &= \frac {i}{\kappa } \left ( \overline {K} X - X K^P \right ) U^P_s, \end{align}

Define $X_2$ as in (3.27). Using (3.32), we write $Y_s \overline {U}_{-s} {\overline {P}} = U_{-s} {\overline {P}}$ and $\left (\partial _s Y_s\right ) \overline {U}_{-s} {\overline {P}} = U_{-s} X^*$ , where recall $X = \frac {i}{\kappa } {\overline {P}} T P$ and so $X^* = - \frac {i}{\kappa } P T {\overline {P}}$ . Using these relations, we can re-write the right-hand side of (3.33) as (3.26).

Finally, it remains to show the second part of (3.27). Using that ${\overline {R}}\,{\overline {P}} = {\overline {R}} = {\overline {P}}\,{\overline {R}}$ and $[H_{\mathrm {bo}} - E, {\overline {P}}] = 0$ , we can write

(3.35) \begin{align} S &= {\overline {R}} \left ((H_{\mathrm {bo}}- E) {\overline {P}} K - K {\overline {P}} (H_{\mathrm {bo}} - E) \right ) {\overline {R}} \nonumber \\ &= {\overline {R}} \left ( (H_{\mathrm {bo}} - E) K - K (H_{\mathrm {bo}} - E) \right ) {\overline {R}} \nonumber \\ &= {\overline {R}} [H_{\mathrm {bo}} - E , K] {\overline {R}}. \end{align}

Using $K = E + T$ and the fact that $E$ commutes with $H_{\mathrm {bo}}$ and itself, we find $[H_{\mathrm {bo}} - E , K] = [H_{\mathrm {bo}} - E , T]$ . This concludes the proof.

The last step is to estimate the operators appearing on the right-hand side of (3.26). The basic building blocks are the smoothness of fibres of $E$ , $P$ , $\psi _\circ$ and $\overline {R}$ . These are standard results in adiabatic perturbation theory and we sketch them in Appendix A. Building on these results, we compute and estimate the various commutators that we encounter in $S$ and $X_2$ , those estimates are less standard and we delegate them to Appendix C.

Lemma 3.4. Let Assumptions [A1]–[A4] hold. Then, we have

(3.36) \begin{equation} S = O_{{\mathcal{L}}_{s+1,s}}(\kappa ), \quad X_2 = O_{{\mathcal{L}}_{2,0}}(1), \end{equation}

and estimating the right-hand side of ( 3.26 ), we obtain

(3.37) \begin{equation} X_t = O_{{\mathcal{L}}_{2,0}}(\kappa \langle t \rangle ^3). \end{equation}

Proof of Lemma 3.4. Our first step is to show the estimate for $S$ in (3.36). In Lemma C.5, we show that $[H_{\mathrm {bo}}, T]$ and $[E,T]$ are $O_{{\mathcal{L}}_{s+1,s}}(\kappa )$ . Writing $S = {\overline {R}} \left ([H_{\mathrm {bo}}, T] - [E , T] \right ) {\overline {R}}$ and using

(3.38) \begin{equation} {\overline {R}} = O_{{\mathcal{L}}_{s,s}}(1), \end{equation}

from Lemma A.3, the desired estimate follows.

Now, we estimate the terms on the right-hand side of (3.33). Using ${\overline {P}} P = 0$ , we have $X = \frac {i}{\kappa } {\overline {P}} [T, P] P$ and $X^* = \frac {i}{\kappa } P [T, P] {\overline {P}}$ . Then by Lemma C.5 part a), $X$ and $X^*$ are both $O_{{\mathcal{L}}_{1,0}}(1)$ . Making use of the propagator estimates

(3.39) \begin{equation} U_t = O_{{\mathcal{L}}_{s,s}}(\langle t \rangle ^s), \quad U^P_t = O_{{\mathcal{L}}_{s,s}}(\langle t \rangle ^s), \end{equation}

(see Appendix D) and (3.6), this shows that the first and second terms on the right-hand side of (3.26) are $O_{{\mathcal{L}}_{2,0}}(\kappa \langle t \rangle ^3)$ . In order to estimate the third term, i.e. $X_2$ , we use the orthogonality of $P$ and $\overline {P}$ combined with Lemma C.3, whose prove can be found in the Appendix. In particular, applying Lemma C.3 with $X^\circ = \frac {i}{\kappa } [T,P]$ , then $X = {\overline {P}} X^\circ P$ and hence

(3.40) \begin{align} X_2 &= \frac {i}{\kappa } S {\overline {P}} X^\circ P + \frac {i}{\kappa } {\overline {R}} \left [ X^\circ , [K,P] - K \right ] P U^P_s = O_{{\mathcal{L}}_{2,0}}(1), \end{align}

where the estimate is shown in Lemma C.6. In combination with the estimates (3.39), this shows that the third term on the right-hand side of (3.26) is also $O_{{\mathcal{L}}_{2,0}}(\kappa \langle t \rangle ^3)$ . This gives $X_t = O_{{\mathcal{L}}_{2,0}}(\kappa \langle t \rangle ^3)$ .

3.3. Conclusion of the proof

We now conclude the proof of Theorem1.1.

Proof of Theorem 1.1. Let $f$ as in the statement. Then, $f = P\Psi$ and thanks to identity (3.12) in Lemma 3.2, we may write

\begin{equation*} U_t(\psi _\circ\, f ) - \psi _\circ e^{i h_{\mathrm {eff}} t/k } f = (U_t - U_t^P )P\Psi . \end{equation*}

Next, using Lemma 3.3, we have

\begin{equation*} (U_t - U_t^P )P\Psi = U_t X_t \Psi , \end{equation*}

for $X_t$ defined in Equation (3.25). Using Lemma 3.4, we deduce that there exists a constant $C\gt 0$ such that for every $\Psi \in {\mathcal{H}}_{\mathrm {el}} H^{2, \kappa }_{\mathrm {nucl}}$ ,

(3.41) \begin{equation} \Vert X_t \Psi \Vert _{H^{2, \kappa }_{\mathrm {nucl}} {\mathcal{H}}_{\mathrm {el}}} \leq C \kappa \langle t \rangle ^3 \Vert \Psi \Vert _{{\mathcal{H}}_{\mathrm {el}} {\mathcal{H}}_{\mathrm {nucl}}}. \end{equation}

Then, using the propagator estimate (D.2) of Theorem D.1, we get

\begin{align*} \Vert U_t(\psi _\circ\, f ) - \psi _\circ e^{i h_{\mathrm {eff}} t/k } f \Vert _{{\mathcal{H}}_{\mathrm {el}} H^{2, \kappa }_{\mathrm {nucl}}} &= \Vert U_t X_t \Psi \Vert _{{\mathcal{H}}_{\mathrm {el}} H^{2, \kappa }_{\mathrm {nucl}}} \\ &\lesssim \Vert X_t \Psi \Vert _{{\mathcal{H}}_{\mathrm {el}} H^{2, \kappa }_{\mathrm {nucl}}} \\ &\lesssim C \kappa \langle t \rangle ^3 \Vert \Psi \Vert _{{\mathcal{H}}_{\mathrm {el}} {\mathcal{H}}_{\mathrm {nucl}}}. \end{align*}

This implies the result.Footnote ⁵

4.1. Proof of Theorem1.2, part (a)

We begin with the following lemma that will be useful in the proof.

Lemma 4.1. Let $\Psi _0 \in {\mathcal{H}}_{\mathrm {el}}$ and let $\Psi$ satisfy ( 1.14 ). The projections $\phi (t) \;:\!=\; P \Psi (t)$ and $ \overline {\phi }(t) \;:\!=\; \overline {P} \Psi (t)$ satisfy

(4.1) \begin{equation} \left \{\begin{array}{l@{\quad}l} i \kappa \partial _t \phi = H^P \phi + P T \overline {\phi }, & \phi (0) = P\Psi _0, \\[3pt] i \kappa \partial _t \overline {\phi } = \overline {H}\, \overline {\phi } + \overline {P} T \phi , & \overline {\phi }(0) = \overline {P}\Psi _0. \end{array} \right. \end{equation}

Proof. We apply the projection $P$ to (1.14). As $P$ commutes with $\partial _t$ , we get

(4.2) \begin{equation} i \kappa \partial _t \phi = P H_{\kappa } \Psi . \end{equation}

Next, we compute

(4.3) \begin{align} P H_{\kappa } \psi = P H_{\kappa } (\phi + \overline {\phi } ) &= P H_{\kappa } \phi + P (T + H_{\mathrm {bo}}) \overline {\phi } \nonumber \\ & = (P H_{\kappa } P) P \phi + PT \overline {\phi } + P H_{\mathrm {bo}} \overline {P} \psi \nonumber \\ & = H^P \phi + PT \overline {\phi }, \end{align}

where we have used that $P^2 = P$ and $P H_{\mathrm {bo}} {\overline {P}} = 0$ since $P$ commutes with $H_{\mathrm {bo}}$ . As a result, we get

(4.4) \begin{equation} i \kappa \partial _t \phi = H^P \phi + PT \overline {\phi } \end{equation}

and the first equation in (4.1) follows. Next, by applying the projection $\overline {P}$ to (1.14) and using that $P H_{\mathrm {bo}} {\overline {P}} = 0 = {\overline {P}} H_{\mathrm {bo}} P$ , so that $P H {\overline {P}} = P T {\overline {P}}$ and ${\overline {P}} H P = {\overline {P}} T P$ , arguing as before yields the second equation in (4.1).

We are now in position to prove the result.

Proof of Theorem 1.2 (a). Step 1: Showing that solutions of (1.14) imply solutions of (1.30). Let $f_0 \in {\mathcal{H}}_{\mathrm {nucl}}$ and $\Psi _0 = \psi _0 f_0$ be as in the statement and let $\Psi$ be the solution to (1.14) associated to the initial datum $\Psi _0$ given above. Then, thanks to Lemma4.1, $\phi (t) = P \Psi (t)$ and $\overline {\phi }(t) = \overline {P} \Psi (t)$ satisfy (4.1). Moreover, if we associate to $\Psi$ the path $t \mapsto f(t)$ defined by (1.15), we may write further

(4.5) \begin{equation} \phi (y,t) = \psi _\circ (y) f(y,t). \end{equation}

In particular, by construction, $\overline {\phi }(0) = \overline {P} \Psi (0) = \overline {P}( \psi _0 f_0 ) = 0$ . Hence, $\overline {\phi }$ satisfies the inhomogeneous Cauchy problem

(4.6) \begin{equation} i \kappa \partial _t \overline {\phi } = \overline {H}\, \overline {\phi } + \overline {P} T \phi , \quad \phi (0) = 0. \end{equation}

Using the semigroup $\overline {U}_t = \epsilon ^{-i\overline {H} t/\kappa }$ , $t\in {\mathbb{R}}$ and the Duhamel principle, we may write

(4.7) \begin{equation} \overline {\phi }(t) = - \frac {i}{ \kappa } \int _0^t \overline {U}_{t-s} \overline {P} T \phi (s)\, \text{d}s. \end{equation}

Next, substituting (4.7) into (4.1), we obtain

(4.8) \begin{equation} i \kappa \partial _t \phi = H^P \phi + {\mathcal{K}} \phi , \end{equation}

where the operator $\mathcal{K}$ is defined as

(4.9) \begin{equation} ({\mathcal{K}} \phi )(t) = - \frac {i}{ \kappa } P T \overline {P} \int _0^t \overline {U}_{t-s} \overline {P} T P \phi (s)\, \text{d}s. \end{equation}

Now, we want to show that the $\mathcal{K}$ can be rewritten as

(4.10) \begin{equation} ({\mathcal{K}} \phi )(y,t) = \psi _\circ (y) w^\kappa [f](y,t) \end{equation}

where $f$ is given in (1.15) and $w$ is given by (1.32). Recalling the direct integral representation of $P$ , we find

(4.11) \begin{align} ({\mathcal{K}} \phi )(y,t) &= - \frac {i}{\kappa } \psi _\circ (y) \langle \psi _\circ (y), P T {\overline {P}} \int _0^t \overline {U}_{t-s} \overline {P} T P \psi _\circ (y) f(s)\, \text{d}s \rangle _{{\mathcal{H}}_{\mathrm {el}}} \nonumber \\ &= \psi _\circ (y) \left ( - \frac {i}{\kappa } \int _0^t \langle \psi _\circ (y), P T {\overline {P}} \ \overline {U}_{t-s} \overline {P} T P \psi _\circ (y) \rangle _{{\mathcal{H}}_{\mathrm {el}}} f(s)\, \text{d}s \right ), \end{align}

so (1.32) follows. Using (3.12), (4.8), (4.10), and dropping $\psi _\circ$ from (4.8), we obtain (1.30).

Step 2: Showing that solutions of (1.30 imply solutions of (1.14). Let $f$ satisfy (1.30) with initial conditions $f_0$ , and let $\Psi = Q_P f$ with $\overline {\phi }_0 = 0$ . Then, (1.29) shows that $\Psi |_{t=0} = \psi _\circ (y) f_0(y)$ . Now, we show that $\Psi$ satisfies (1.14). Using (1.30), we compute

(4.12) \begin{align} i \kappa \partial _t \Psi &= \psi _\circ (i \kappa \partial _t f) + \overline {H} \Psi _1 + \overline {P} T \psi _\circ\, f \nonumber \\ &= \psi _\circ \left ( E + T + \kappa ^2 v\right ) f + \psi _\circ w^\kappa [f] + \overline {H} \Psi _1 + \overline {P} T \psi _\circ\, f, \end{align}

where

\begin{equation*} \Psi _1 \;:\!=\; - \frac {i}{ \kappa } \int _0^t \overline {U}_{t-s} \overline {P} T \psi _\circ\, f\, \text{d}s. \end{equation*}

Now, we identify the terms on the right-hand side of (4.12). By (3.12) and the relation $P \Psi = \psi _\circ\, f$ , we have

(4.13) \begin{equation} \psi _\circ \left ( E + T + \kappa ^2 v \right ) f = H^P \Psi . \end{equation}

Next, note that $P H \overline {P} = P T \overline {P}$ and hence, using (1.32) and the definition of $P$ in (1.24), we find that

(4.14) \begin{equation} \psi _\circ w^\kappa [f] = P T \Psi _1 = P H \overline {P} \Psi . \end{equation}

Similarly $\overline {P} H P = \overline {P} T P$ , and so, by $\psi _\circ\, f = P \Psi$ , we have

(4.15) \begin{equation} \overline {P} T \psi _\circ\, f = \overline {P} H P \Psi . \end{equation}

Using (4.13), (4.14) and (4.15) we see that the right-hand side of (4.12) equals $(P HP + P H \overline {P} + \overline {P} H \overline {P} + \overline {P} H P)\Psi = H \Psi$ , and hence $\Psi = Q_P f$ satisfies the Schrödinger equation (1.7) with initial condition $\Psi _0(x,y) = \psi _\circ (x,y) f_0(y)$ .

4.2. Main ideas of the proof of Theorem1.2, part (b)

In this subsection, we explain the main idea behind the proof of part (b) of Theorem1.2. In subsequent subsections, we will close the argument and prove Theorem1.2 (b) while relying on Lemmas 3.1, A.2 and A.3, as well as results from Appendices C and D.

Our goal is to decompose the electronic feedback operator $w^{\kappa }$ defined in (1.32) into a local (Markov) term and a higher order (in $\kappa$ ) non-local one. To this end, assume that $f$ obeys the Schrödinger equation (1.16) with the effective nuclear Hamiltonian (1.31). Recall in previous sections we used $X = \frac {i}{\kappa } {\overline {P}} T P$ and its adjoint $X^* = - \frac {i}{\kappa } P T {\overline {P}}$ . We re-write (1.32) as

(4.16) \begin{align} w^\kappa [f](t) &= - i \kappa \langle \psi _\circ , X^* \left (A_t^X f\right )(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \end{align}

where we have introduced the operator

(4.17) \begin{equation} \left (A_t^Xf\right )\!(t)= \int _0^t \overline {U}_{t-r} X \psi _\circ\, f(r)\, \text{d}r. \end{equation}

Now, similar to the treatment of $X_t$ in Theorems1.1, we expand $A_t f$ further by using the NAIP. Then, each integration by parts produces a number of local and non-local terms involving commutators, with each commutator contributing $O(\kappa )$ , as shown in Section C.

4.3. Expansion of the electronic feedback operator $\boldsymbol{w}^{\kappa }$

In this section, we consider integrals of the form

(4.18) \begin{equation} \left (A_t^Y f \right )(t) = \int _0^t \overline {U}_{t-r} Y \psi _\circ\, f(r)\, \text{d}r, \end{equation}

where $Y\;:\; {\mathrm {Ran}}\, P \to {\mathrm {Ran}}\, {\overline {P}}$ is a given operator and study the expansion of such $A_t^Y f$ in terms of other operators of the same kind. Lemma 4.2 is one of the cornerstones of our proof of Theorem1.2, part (b) and essentially, it is an induction step for the expansion of $A_t^X f$ .

Lemma 4.2. Let $Y\;:\; {\mathrm {Ran}}\, P \to {\mathrm {Ran}}\, {\overline {P}}$ . For all $t \in \mathbb{R}$ , and for all $f$ solutions of ( 1.30 ), $A_t^{Y}f$ as given in ( 4.18 ) has the following expansion:

(4.19) \begin{align} \left (A_t^{Y} f\right )(t) &= - i \kappa \left [ {\overline {R}} Y \psi _\circ\, f(t) - \overline {U}_t {\overline {R}} Y \psi _\circ\, f_0 \right ] - i \kappa \left (A_t^{{\overline {R}} Y X^*} A^X_{(\cdot)}\, f \right )(t) \nonumber \\ &\quad + i \kappa \left (A^{W[Y]}_t f\right )(t), \end{align}

where the operator $Y_{j+1}$ is defined recursively through the recursion relation ( 4.21 ), i.e.

(4.20) \begin{equation} W[Y] = \frac {i}{\kappa } S Y + \frac {i}{\kappa } {\overline {R}} \left ( \overline {K} Y - Y K^P \right )P, \end{equation}

where recall $S$ is given by ( 3.27 ). Notice that $W[Y]\;:\;{\mathrm {Ran}}\, P \to {\mathrm {Ran}}\, {\overline {P}}$ .

Repeated applications of the operation $W[\!\cdot\! ]$ on $X = \frac {i}{\kappa } {\overline {P}} T P$ generates the recursive relation

(4.21) \begin{equation} \begin{cases} X_1 = \frac {i}{\kappa } {\overline {P}} T P, \\[3pt] X_j = \frac {i}{\kappa } S X_{j-1} + \frac {i}{\kappa } {\overline {R}} \left (K {\overline {P}} X_{j-1} - X_{j-1} P K \right ) P, \quad j \geqslant 2, \end{cases} \end{equation}

defining the family $\{X_j\}_{j \geqslant 1}$ of differential operators.

Proof of Lemma 4.2. We apply Lemma 2.2 to integrate $A^Y_t f$ by parts, with $G_s = 1$ , $A = \frac {i}{\kappa } \overline {H}$ , $B = \frac {i}{\kappa } \overline {K}$ and $F_s = Y \psi _\circ\, f(s)$ . Then, we apply $\overline {U}_t$ to the right-hand side of (2.5) to obtain

(4.22) \begin{align} \left (A_t^{Y} f\right )(t) &= \overline {U}_{t-r} {\overline {R}} Y \psi _\circ\, f(r) \bigg |_{r=0}^{r=t} + \int _0^t \overline {U}_{t-r} S Y \psi _\circ\, f(r)\, \text{d}r \nonumber \\ &\quad - \int _0^t \overline {U}_{t-r} {\overline {R}} C_r\, \text{d}r, \end{align}

where as before $S = {\overline {R}} [\overline {H}_{\mathrm {bo}} - \overline {E}, \overline {K}] {\overline {R}}$ and ${\overline {R}} = {\overline {P}} (\overline {H}_{\mathrm {bo}} - \overline {E})^{-1}{\overline {P}}$ , and

(4.23) \begin{align} C_r &\;:\!=\; \frac {i}{\kappa } \overline {K} Y \psi _\circ\, f(r) + \frac {\partial }{\partial r} \left (Y \psi _\circ\, f(r) \right )\!. \end{align}

Computing the time-derivative in $C_r$ and using that $f(r)$ solves (1.30), we write

(4.24) \begin{align} - i \kappa C_r &= \overline {K} Y \psi _\circ\, f(r) - Y \psi _\circ (i \kappa \partial _{r} f(r)) \nonumber \\ &= \overline {K} Y \psi _\circ\, f(r) - Y \psi _\circ \left (K + \kappa ^2 v\right ) f(r) - Y \psi _\circ w^\kappa [f](r). \end{align}

Using the identity (3.11) and writing

(4.25) \begin{equation} \psi _\circ w^\kappa [f](r) = - i \kappa X^* \left ( A_r^X f \right )(r), \end{equation}

see (4.16), on the right-hand side of (4.24), we obtain

(4.26) \begin{align} - i \kappa C_r &= \left ( \overline {K} Y - Y K^P \right ) \psi _\circ\, f(r) - i \kappa Y X^* \left ( A_r^X f \right )(r). \end{align}

Substituting (4.26) into (4.22) and inserting $W[Y]$ using (4.20), we can identify all four terms in (4.19).

Recall that $X$ and $X^*$ are both $O_{{\mathcal{L}}_{s+1,s}}(1)$ . Hence, a direct estimate on $\left (A^{X}_t f \right )(t)$ would be $O(1)$ , but estimating after expanding using (4.19) gives us an estimate of $O(\kappa )$ . In fact, we obtain

(4.27) \begin{align} \left (A_t^{X} f \right )(t) &= - i \kappa \left [ {\overline {R}} X \psi _\circ\, f(t) - \overline {U}_t {\overline {R}} X \psi _\circ\, f_0 \right ] - i \kappa \left (A_t^{{\overline {R}} X X^*} A^X_{(\cdot)}\, f \right )(t) \nonumber \\ &\quad + i \kappa \left (A^{X_2}_t f\right )(t), \end{align}

where we have two local terms and two non-local terms. The non-local terms can be expanded further by iterating the expansion to obtain higher-order local corrections and non-local remainders. This is in fact necessary in order to obtain a higher-order remainder, as recall by Proposition 4.7, $X_2=O_{{\mathcal{L}}_{s+2,s}}(1)$ in $\kappa$ and hence both non-local terms in (4.27) are of the same order as the local terms, $O(\kappa )$ . This is the content of the Lemma 4.3.

Lemma 4.3. We can expand $w^\kappa [f](t)$ into the following form

(4.28) \begin{align} w^\kappa [f](t) = (\!-\!i\kappa )^{2} \left ( w_1 f(t) - \tilde w_1(t) f_0 \right ) + (\!-\!i\kappa )^{3} w^{\kappa }_{2}[f](t), \end{align}

where

(4.29) \begin{align} w_1 f(t) &= \langle \psi _\circ , X^* {\overline {R}} X \psi _\circ\, f(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \end{align}

(4.30) \begin{align} \tilde w_1(t) f_0 &= \langle \psi _\circ , X^* \overline {U}_t {\overline {R}} X \psi _\circ\, f_0 \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \end{align}

(4.31) \begin{align} w^{\kappa }_{2}[f](t) &= - \left (w_2 f(t) - \tilde w_2(t) f_0 \right ) \nonumber \\ &\quad + \int _0^t \langle \psi _\circ , X^* \overline {U}_{t-r} \left (X_3 + {\overline {R}} X X^* {\overline {R}} X \right ) \psi _\circ\, f(r) \rangle _{{\mathcal{H}}_{\mathrm {el}}}\, \text{d}r \nonumber \\ &\quad + \int _0^t \int _0^r \langle \psi _\circ , X^* \overline {U}_{t-r} {\overline {R}} X_2 X^* \overline {U}_{r-s} X \psi _\circ\, f(s) \rangle _{{\mathcal{H}}_{\mathrm {el}}}\, \text{d}s\, \text{d}r \nonumber \\ &\quad + \int _0^t \int _0^r \langle \psi _\circ , X^* \overline {U}_{t-r} {\overline {R}} X X^* \overline {U}_{r-s} X_2 \psi _\circ\, f(s) \rangle _{{\mathcal{H}}_{\mathrm {el}}}\, \text{d}s\, \text{d}r \nonumber \\ &\quad + \int _0^t \int _0^r \int _0^s \langle \psi _\circ , X^* \overline {U}_{t-r} {\overline {R}} X X^* \overline {U}_{r-s} {\overline {R}} X X^* \overline {U}_{s-q} X \psi _\circ\, f(q) \rangle _{{\mathcal{H}}_{\mathrm {el}}} dq\, \text{d}s\, \text{d}r, \\[8pt] \nonumber\end{align}

and

(4.32) \begin{align} w_2f(t) &= \langle \psi _\circ , X^* {\overline {R}} X_2 \psi _\circ\, f(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \end{align}

(4.33) \begin{align} \tilde w_2(t) f_0 &= \langle \psi _\circ , X^* \overline {U}_t {\overline {R}} X_2 \psi _\circ\, f_0 \rangle _{{\mathcal{H}}_{\mathrm {el}}} \nonumber \\ &\quad - \int _0^t \langle \psi _\circ , X^* \overline {U}_{t-r} {\overline {R}} X X^* \overline {U}_{r} {\overline {R}} X \psi _\circ\, f_0 \rangle _{{\mathcal{H}}_{\mathrm {el}}}\, \text{d}r.\\[8pt] \nonumber \end{align}

In particular, recall that $X = \frac {i}{\kappa } {\overline {P}} [T, P] P$ , $X^* = \frac {i}{\kappa } P[T, P] {\overline {P}}$ , and, c.f. ( 4.21 ),

(4.34) \begin{align} X_2 &= -\frac {1}{\kappa ^2} S {\overline {P}} [T, P] P - \frac {1}{\kappa ^2} {\overline {R}} \left [[T, P], [T, P] - K \right ] P, \end{align}

(4.35) \begin{align} X_3 &= -\frac {i}{\kappa ^3} S^2 {\overline {P}} [T, P] P - \frac {i}{\kappa ^3} {\overline {R}} \left [S, [T, P] - K \right ] [T, P] P \nonumber \\ &\quad - \frac {i}{\kappa ^3} {\overline {R}} S \left [[T, P], [T, P] - K \right ] P - \frac {i}{\kappa ^3} S {\overline {R}} \left [[T, P], [T, P] - K \right ] P \nonumber \\ &\quad - \frac {i}{\kappa ^3} {\overline {R}} \left [ {\overline {R}}, [T, P] - K \right ] \left [[T, P], [T, P] - K \right ] P \nonumber \\ &\quad - \frac {i}{\kappa ^3} {\overline {R}}^2 \left [ \left [[T, P], [T, P] - K \right ], [T, P] - K \right ]. \end{align}

Proof of Lemma 4.3. We begin with the expansion (4.27) with $n=1$ and expand the two non-local terms. For the term $\left (A_t^{{\overline {R}} X X^*} A^X_{(\cdot)}\, f \right )(t)$ , we use Lemma 4.2 with $Y = X$ to expand $A^X_{(\cdot)}\, f$ and obtain

(4.36) \begin{align} \left (A_t^{{\overline {R}} X X^*} A^X_{(\cdot)}\, f \right )(t) &= (\!-\!i\kappa ) \left ( A_t^{{\overline {R}} X X^* {\overline {R}} X} f\right )(t) \nonumber \\ &\quad - (\!-\!i\kappa ) \left ( A_t^{{\overline {R}} X X^* } \overline {U}_{(\cdot)}\right )(t) {\overline {R}} X \psi _\circ\, f_0 \nonumber \\ &\quad - (\!-\!i\kappa ) \left ( A_t^{{\overline {R}} X X^* } A_{(\cdot)}^{X_2}\, f \right )(t) \nonumber \\ &\quad + (\!-\!i\kappa ) \left ( A_t^{{\overline {R}} X X^*} A_{(\cdot)}^{{\overline {R}} X X^*} A_{(\cdot)}^{X}\, f \right )(t). \end{align}

Expanding $\left (A^{X_2}_t f\right )(t)$ using Lemma 4.2 with $Y = X_2$ yields

(4.37) \begin{align} \left (A^{X_2}_t f\right )(t) &= - (\!-\!i\kappa ) \left ({\overline {R}} X_2 \psi _\circ\, f(t) - \overline {U}_t {\overline {R}} X_2 \psi _\circ\, f_0 \right ) \nonumber \\ &\quad + (\!-\!i\kappa ) \left (A_t^{{\overline {R}} X_2 X^*} A^X_{(\cdot)}\, f \right )(t) - (\!-\!i\kappa ) \left (A_t^{X_3} f \right )(t). \end{align}

Using these expansions and plugging them back into (4.27), we obtain an expansion of the form

(4.38) \begin{align} \left (A_t^{X} f \right )(t) &= - i \kappa \left ( {\overline {R}} X \psi _\circ\, f(t) - \overline {U}_t {\overline {R}} X \psi _\circ\, f_0 \right ) \nonumber \\ &\quad - (\!-\!i\kappa )^2 \left ({\overline {R}} X_2 \psi _\circ\, f(t) - \overline {U}_t {\overline {R}} X_2 \psi _\circ\, f_0 \right ) \nonumber \\ &\quad - (\!-\!i\kappa )^2 \left ( A_t^{{\overline {R}} X X^* } \overline {U}_{(\cdot)}\right )(t) {\overline {R}} X \psi _\circ\, f_0 \nonumber \\ &\quad + (\!-\!i\kappa )^2 \left (A_t^{X_3 + {\overline {R}} X X^* {\overline {R}} X} f \right )(t) \nonumber \\ &\quad + (\!-\!i\kappa )^2 \left ( A_t^{{\overline {R}} X X^* } A_{(\cdot)}^{X_2}\, f \right )(t) + (\!-\!i\kappa )^2 \left (A_t^{{\overline {R}} X_2 X^*} A^X_{(\cdot)}\, f \right )(t) \nonumber \\ &\quad + (\!-\!i\kappa )^2 \left ( A_t^{{\overline {R}} X X^*} A_{(\cdot)}^{{\overline {R}} X X^*} A_{(\cdot)}^{X}\, f \right )(t). \end{align}

Now, we can write the expansion for $w^\kappa [f](t)$ by inserting (4.38) into the expression (4.16),

(4.39) \begin{align} w^\kappa [f](t) = (\!-\!i\kappa )^{2} \left ( w_1 f(t) - \tilde w_1(t) f_0 \right ) + (\!-\!i\kappa )^{3} w^{\kappa }_{2}[f](t), \end{align}

where

(4.40) \begin{align} w_1 f(t) &= \langle \psi _\circ , X^* {\overline {R}} X \psi _\circ\, f(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \end{align}

(4.41) \begin{align} \tilde w_1(t) f_0 &= \langle \psi _\circ , X^* \overline {U}_t {\overline {R}} X \psi _\circ\, f_0 \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \end{align}

(4.42) \begin{align} w^{\kappa }_{2}[f](t) &= - \left (\langle \psi _\circ , X^* {\overline {R}} X_2 \psi _\circ\, f(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}} - \langle \psi _\circ , X^* \overline {U}_t {\overline {R}} X_2 \psi _\circ\, f_0 \rangle _{{\mathcal{H}}_{\mathrm {el}}} \right ) \nonumber \\ &\quad - \langle \psi _\circ , X^* \left (A^{{\overline {R}} X X^*}_t \overline {U}_{(\cdot)} \right )(t) {\overline {R}} X \psi _\circ\, f_0 \rangle _{{\mathcal{H}}_{\mathrm {el}}} \nonumber \\ &\quad + \langle \psi _\circ , X^* \left (A_t^{X_3 + {\overline {R}} X X^* {\overline {R}} X} f \right )(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}} \nonumber \\ &\quad + \langle \psi _\circ , X^* \left (A_t^{{\overline {R}} X_2 X^*} A_{(\cdot)}^{X}\, f \right )(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}} \nonumber \\ &\quad + \langle \psi _\circ , X^* \left (A_t^{{\overline {R}} X X^*} A_{(\cdot)}^{X_2}\, f \right )(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}} \nonumber \\ &\quad + \langle \psi _\circ , X^* \left (A_t^{{\overline {R}} X X^*} A_{(\cdot)}^{{\overline {R}} X X^*} A_{(\cdot)}^{X}\, f \right )(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}.\\[8pt] \nonumber \end{align}

Using the definition (4.18) of the operators $A^{Y}_t f$ and identifying $w_2f(t)$ and $\tilde w_2(t) f_0$ via (4.32) and (4.33), we obtain (4.28)–(4.33). The expansions (4.34) and (4.35) follow directly from the recursive relation (4.21).

Next, we would like to obtain a closed-form expression of $\left (A^{X}_t f \right )(t)$ expanded to arbitrary order. It is clear that one can always expand using Lemma 4.2 to arbitrary order, and in the following proposition, we aim to at least characterise this expansion. However, the recursion relations are too complicated to characterise explicitly.

Proposition 4.4. Suppose Assumptions [A1]–[A4] hold. For all $t \in \mathbb{R}$ , and for all $f$ solutions of ( 1.30 ), $A_t^X$ as given in ( 4.17 ) admits an expansion of the following form:

(4.43) \begin{equation} \left (A_t^{X} f\right )(t) = \sum _{j=1}^n - (\!-\!i\kappa )^j \left [ L_j \psi _\circ\, f(t) - \tilde L_j(t) \psi _\circ\, f_0 \right ] + (\!-\!i\kappa )^n {\mathcal{R}}_n [f](t), \end{equation}

where $L_j$ and $L_j(t)$ are local operators satisfying the recursion relation below, and the remainder ${\mathcal{R}}_n$ is given by

(4.44) \begin{align} {\mathcal{R}}_n [f](t) &= \left (A^{Y_n}_t f \right ) (t) + \sum _{j=2}^n \sum _{l=1}^{N(n,j)} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} A^{Y_{n, j}^{l,j-2}}_{(\cdot)} \dots A^{Y_{n, j}^{l,1}}_{(\cdot)}\, f \right )(t) \nonumber \\ &\quad + \left ( A^{{\overline {R}} X X^*}_t \left (\prod _{1}^{n-1} A^{{\overline {R}} X X^*}_{(\cdot)}\right ) A^{X}_{(\cdot)}\, f \right )(t), \end{align}

for local time-independent differential operators $Y_{n, j}^{l,k}$ . The $n=1$ case is given by ( 4.27 ), in which case we have $L_1 = {\overline {R}} X$ , $\tilde L_1(t) = \overline {U}_t {\overline {R}} X$ and $Y_1 = X_2$ , and the $n=2$ case is given by ( 4.38 ).

For $n \geqslant 2$ , we describe the following recursive relations. $N(n,j)$ is a sequence of integers counting the number of respective terms, given by the recursive relation $N(n,1) = 1$ , $N(n,n+1) = 1$ , $N(n, j) = 0$ for $j \geqslant n+2$ , and

(4.45) \begin{equation} N(n+1,j) = N(n, j-1) + N(n,j) + N(n, j+1). \end{equation}

For all values of $n$ and $j$ the operators $Y_n$ and $Y_{n,j}^{l,k}$ are differential operators belonging to the class ${\mathcal{C}}^p$ for some $p \geqslant 1$ . Furthermore, we have the following recursion relations:

(4.46) \begin{equation} L_{n+1} = {\overline {R}} Y_n, \quad n \geqslant 0, \end{equation}

where

(4.47) \begin{equation} Y_{n} = - W(Y_{n-1}) + \sum _{l=1}^{N(n-1,2)} Y_{n,2}^{l,2} {\overline {R}} Y_{n,2}^{l,1}, \quad Y_1 \;:\!=\; X, \end{equation}

with $W(Y_{n-1}) = \frac {i}{\kappa } S Y_{n-1} + \frac {i}{\kappa } {\overline {R}} \left (\overline {K} Y_{n-1} - Y_{n-1} K^P \right )P$ . Also, $\tilde L_{1}(t) = \overline {U}_t {\overline {R}} X$ and for $n \geqslant 1$ ,

(4.48) \begin{align} \tilde L_{n+1}(t) &= \overline {U}_t {\overline {R}} Y_n - \sum _{j=2}^n \sum _{l=1}^{N(n,j)} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} A^{Y_{n, j}^{l,j-2}}_{(\cdot)} \dots \overline {U}_{(\cdot)} \right )(t) Y_{n, j}^{l,1} \nonumber \\ &\quad - \left ( A^{{\overline {R}} X X^*}_t \left (\prod _{1}^{n-1} A^{{\overline {R}} X X^*}_{(\cdot)}\right ) \overline {U}_{(\cdot)} \right )(t) {\overline {R}} X. \end{align}

Proof of Proposition 4.4. For the inductive step, we expand the remainder $({\mathcal{R}}_n f)(t)$ by expanding the innermost $A^{Y}_{(\cdot)}\, f$ term in every term of the form

(4.49) \begin{equation} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} A^{Y_{n, j}^{l,1}}_{(\cdot)}\, f \right )(t), \end{equation}

using the expansion (4.19) with $Y = Y_{n, j}^{l,1}$ to obtain

(4.50) \begin{align} & \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} A^{Y_{n, j}^{l,1}}_{(\cdot)}\, f \right )(t) \nonumber \\[4pt] &\quad = (\!-\!i\kappa ) \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2} {\overline {R}} Y_{n,j}^{l,1}}_{(\cdot)}\, f \right )(t) \nonumber \\[4pt] &\qquad - (\!-\!i\kappa ) \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} \overline {U}_{(\cdot)} \right )(t) {\overline {R}} Y_{n,j}^{l,1} \psi _\circ\, f_0 \nonumber \\[4pt] &\qquad - (\!-\!i\kappa ) \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} A^{W(Y_{n, j}^{l,1})}_{(\cdot)}\, f \right )(t) \nonumber \\[4pt] &\qquad + (\!-\!i \kappa ) \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} A^{{\overline {R}} Y_{n, j}^{l,1} X^*}_{(\cdot)} A^{X}_{(\cdot)}\, f \right )(t). \end{align}

Here, we have one non-local term of length $j-1$ acting on $\psi _\circ\, f(t)$ , one local time-dependent term of length $j-1$ acting on $\psi _\circ\, f_0$ , one non-local term of length $j$ acting on $\psi _\circ\, f(t)$ and one non-local term of length $j+1$ acting on $\psi _\circ\, f(t)$ . At every value of $n$ , there will always be one term of length $1$ and one term of length $n+1$ . This yields the recursion relation (4.45) for $N(n,j)$ .

To obtain the recursion relations (4.46)–(4.48), we expand the terms of length $j=2$ and $j=1$ in the same manner as (4.50) and collect the like terms.

Inserting (4.43) into the expression (4.16) for $w^{\kappa }[f](t)$ yields

(4.51) \begin{equation} w^\kappa [f](t) = \sum _{j=1}^{n-1} - (\!-\!i\kappa )^{j+1} \left ( w_j f(t) - \tilde w_j(t) f_0 \right ) + (\!-\!i\kappa )^{n+1} w^{\kappa }_{n}[f](t), \end{equation}

with the local operators $w_j$ and $\tilde w_j(t)$ and remainder $w^{\kappa }_{n}[f](t)$ given by

(4.52) \begin{align} w_j f(t) &= \langle \psi _\circ , X^* L_j \psi _\circ\, f(t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \end{align}

(4.53) \begin{align} \tilde w_j(t) f_0 &= \langle \psi _\circ , X^* \tilde L_j(t) \psi _\circ\, f_0 \rangle _{{\mathcal{H}}_{\mathrm {el}}}, \end{align}

(4.54) \begin{align} w^{\kappa }_{n}[f](t) &= - \left (w_n f(t) - \tilde w_n(t) f_0\right ) + \langle \psi _\circ , X^* {\mathcal{R}}_n [f](t) \rangle _{{\mathcal{H}}_{\mathrm {el}}}.\\[8pt] \nonumber \end{align}

In the next subsection, we estimate the right-hand sides of (4.52)–(4.54).

4.4. Completion of proof of Theorem1.2 (b)

In this section, we estimate (4.51) completing the proof of Theorem1.2 (b). All terms $L_j$ , $\tilde L_j(t)$ and ${\mathcal{R}}_n[f](t)$ can be estimated using the following two Proposition 4.7 and Proposition 4.4.

Finding an explicit expression for the recursive relation (4.21) is too hard. Instead, we will obtain our estimate recursively by first showing

(4.55) \begin{equation} X_j = O_{{\mathcal{L}}_{s+j,s}}(1). \end{equation}

To do so, we introduce ${\mathcal{C}}^{k}$ , the class of differential operators of order $k$ with coefficients of a certain form. Consider an operator formed by linear combinations of compositions of the bounded operators

(4.56) \begin{equation} (\partial _y^{\alpha _1} H_{\mathrm {bo}}), \:(\partial _y^{\alpha _2} P), \:(\partial _y^{\alpha _3} E), \text{ and } {\overline {R}}, \end{equation}

for multi-indices $\alpha _1 \geqslant 1$ , and $\alpha _2, \alpha _3 \geqslant 0$ as permitted by the regularity from Assumption [A1]. Consider operators of the form

(4.57) \begin{equation} A = \sum _{0 \leqslant \left |\alpha \right | \leqslant k} c_\alpha B_{\alpha } D^\alpha = O_{{\mathcal{L}}_{s+k,s}}(1), \end{equation}

where, for all multi-indices $\alpha$ and all $j$ , $c_\alpha$ are possibly $\kappa ^{-1}$ -dependent coefficients and $B_\alpha$ are linear combinations of operators in (4.56) labeled by the multi-index $\alpha$ . We define the following class of operators,

(4.58) \begin{equation} {\mathcal{C}}^{k} \;:\!=\; \left \{ A \text{ operators of the form } (4.57) \right \}. \end{equation}

Proof of Lemma 4.5. We write

(4.59) \begin{align} C_1 C_2 &= \sum _{\substack {\left |\alpha \right | \leqslant k, \\ \left |\beta \right | \leqslant l}} B_\alpha D^\alpha B_\beta D^\beta = \sum _{\substack {\left |\alpha \right | \leqslant k, \\ \left |\beta \right | \leqslant l}} B_\alpha B_\beta D^{\alpha + \beta } + \sum _{\substack {\left |\alpha \right | \leqslant k, \\ \left |\beta \right | \leqslant l}} B_\alpha [D^\alpha , B_\beta ] D^\beta . \end{align}

Clearly, the first term belongs to ${\mathcal{C}}^{k+l}$ , so it remains to show the same for the second term. This follows via a straightforward induction argument, by showing

(4.60) \begin{equation} [D^\alpha , B_\beta ] \in {\mathcal{C}}^{\left |\alpha \right |-1}. \end{equation}

As $B_\beta$ is a linear combination of elements of (4.56), (4.60) follows using Lemma C.5 parts a) and b).

Lemma 4.6. Let $B$ be one of the operators in ( 4.56 ). Then $B = O_{{\mathcal{L}}_{s,s}}(1)$ , $\frac {i}{\kappa } [B, [T,P] - T] \in {\mathcal{C}}^{1}$ , and

(4.61) \begin{equation} \frac {i}{\kappa } [B, [T,P] - T] = O_{{\mathcal{L}}_{s+1,s}}(1). \end{equation}

Proof of Lemma 4.6. Writing $T = D^2$ , then

(4.62) \begin{equation} [T,P] - T = -\kappa ^2 (\Delta _y P) - 2 i \kappa (\nabla _y P) D - D^2. \end{equation}

Hence, using that $B$ is one of the operators in (4.56) and $B = O_{{\mathcal{L}}_{s,s}}(1)$ , we estimate the commutator $[B, [T,P] - T]$ in the manner of (4.61) using Lemma C.5 part a), and in particular, $\frac {i}{\kappa } [B, [T,P] - T] \in {\mathcal{C}}^{1}$ .

We write $X = {\overline {P}} X^\circ P$ , where

(4.63) \begin{equation} X^\circ = \frac {i}{\kappa } [T, P] \in {\mathcal{C}}^{1} \text{ and } X^\circ = O_{{\mathcal{L}}_{s+1,s}}(1) \text{ for } s \leqslant k_A - 1. \end{equation}

In fact, by the recursion relation (4.21), it is clear that $X_j = {\overline {P}} X_j P$ for each $j \geqslant 1$ . Then, we seek to show that there exists a $X_j^\circ \in {\mathcal{C}}^j$ such that $X_j = {\overline {P}} X_j^\circ P$ .

Proof of Proposition 4.7. We argue by induction on $j$ . As the base case is covered by the hypothesis, only the induction step remains. Assume that $X_n^\circ \in {\mathcal{C}}^{n}$ with $X_n = {\overline {P}} X_n^\circ P$ . Then, by the recursion relation (4.21),

(4.64) \begin{equation} X_{n+1} = \frac {i}{\kappa } S X_n + \frac {i}{\kappa } {\overline {R}} \left (K {\overline {P}} X_n - X_n P K \right ) P \end{equation}

and using $X_n = {\overline {P}} X_n^\circ P$ ,

(4.65) \begin{equation} X_{n+1} = \frac {i}{\kappa } S X_n^\circ P + \frac {i}{\kappa } {\overline {R}} \left (K {\overline {P}} X_n^\circ - X_n^\circ P K \right ) P. \end{equation}

Now, applying Lemma C.3 with $X^\circ = X_n^\circ$ yields

(4.66) \begin{equation} X_{n+1} = \frac {i}{\kappa } S X_n^\circ P + \frac {i}{\kappa } {\overline {R}} [X_n^\circ , [K, P] - K] P. \end{equation}

Using $K = T + E$ and $[E, P] = 0$ , then $[K, P] = [T, P]$ so we write $X_{n+1}$ as the sum of three terms,

(4.67) \begin{equation} X_{n+1} = \frac {i}{\kappa } S X_n^\circ P + \frac {i}{\kappa } {\overline {R}} [X_n^\circ , [T, P] - T] P - \frac {i}{\kappa } {\overline {R}} [X_n^\circ , E] P. \end{equation}

Recall that $S = {\overline {R}} [H_{\mathrm {bo}} - E, T] {\overline {R}}$ and using Lemma C.5, it is clear that $S \in {\mathcal{C}}^{1}$ (c.f. (3.36)). Hence, we define

(4.68) \begin{equation} X_{n+1}^\circ = \frac {i}{\kappa } S X_n^\circ + \frac {i}{\kappa } {\overline {R}} [X_n^\circ , [T, P] - T] - \frac {i}{\kappa } {\overline {R}} [X_n^\circ , E]. \end{equation}

Clearly, the class ${\mathcal{C}}^{n+1}$ is closed under linear combinations, so we examine each term individually. Then, by Lemma 4.5, the composition $S X_n^\circ \in {\mathcal{C}}^{n+1}$ .

Now consider the term ${\overline {R}} [X_n^\circ , [T, P] - T]$ . Since $X_n^\circ \in {\mathcal{C}}^{n}$ , we write

(4.69) \begin{align} [X_n^\circ , [T, P] - T] &= \sum _{0 \leqslant \left |\alpha \right | \leqslant n} [B_{\alpha } D^\alpha , [T, P] - T] \nonumber \\ &= \sum _{0 \leqslant \left |\alpha \right | \leqslant n} [B_{\alpha }, [T, P] - T] D^\alpha + B_{\alpha } [D^\alpha , [T, P] - T]. \end{align}

By Lemma 4.6, $[B_{\alpha }, [T,P] - T] \in {\mathcal{C}}^{1}$ . By Lemma C.5 part d), it follows that

(4.70) \begin{equation} [D^\alpha , [T, P] - T] = [D^\alpha , [T, P]] \in {\mathcal{C}}^{\left |\alpha \right |}. \end{equation}

Then by the above, Lemma 4.5, and (4.69),

(4.71) \begin{equation} [X_j^\circ , [T, P] - T] \in {\mathcal{C}}^{\left |\alpha \right |+1}. \end{equation}

The last term is ${\overline {R}}[X_n^\circ , E]$ . Writing

(4.72) \begin{equation} X_n^\circ = \sum _{0 \leqslant \left |\alpha \right | \leqslant n} B_\alpha D^\alpha , \end{equation}

we see that since $B_\alpha$ satisfy (4.56), by LemmaC.1, we have $[B_\alpha , E] = 0$ . Hence,

(4.73) \begin{equation} [X_n^\circ , E] = \sum _{0 \leqslant \left |\alpha \right | \leqslant n} B_\alpha [D^\alpha , E], \end{equation}

and by the Leibnitz Rule Lemma C.2, we have $[D^\alpha , E] \in {\mathcal{C}}^{\left |\alpha \right |-1}$ , so ${\overline {R}} [X_n^\circ , E] \in {\mathcal{C}}^{n-1}$ . This concludes the proof.

Proposition 4.8. Let Assumptions [A1]–[A4] hold. Then, the following estimates hold for $s \geqslant 0$ a positive integer and some constant $C \gt 0$ depending on $s$ . For $n \geqslant 1$ and $t \in {\mathbb{R}}$ ,

(4.74) \begin{equation} L_n = O_{{\mathcal{L}}_{s+n,s}}(1), \quad \tilde L_n (t) = O_{{\mathcal{L}}_{s+2n-1,s}}(e^{Ct}). \end{equation}

For $n \geqslant 1$ and $0 \leqslant t \leqslant \tau$ ,

(4.75) \begin{equation} \lVert {{\mathcal{R}}_n[f](t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim e^{C \tau } \lVert {f}\rVert _{B_\tau ^{s+2n+1}}. \end{equation}

Proof of Proposition 4.8. The $n=1$ case (4.74) follows directly from estimating the terms in (4.27). In particular, we use ${\overline {R}} = O_{{\mathcal{L}}_{s,s}}(1)$ , $X = O_{{\mathcal{L}}_{s+1,s}}(1)$ , and $\overline {U}_t = O_{{\mathcal{L}}_{s,s}}(e^{Ct})$ to see that

(4.76) \begin{align} L_1 = {\overline {R}} X = O_{{\mathcal{L}}_{s+1,s}}(1), \quad L_1 (t) = \overline {U}_t {\overline {R}} X = O_{{\mathcal{L}}_{s+1,s}}(e^{Ct}). \end{align}

To estimate ${\mathcal{R}}_1[f](t)$ , we use in addition to the previous the estimates $X_2 = O_{{\mathcal{L}}_{s+2,s}}(1)$ and $X^* = O_{{\mathcal{L}}_{s+1,s}}(1)$ . In particular, from the definition (4.18) of $A_t^Y f$ , we see that if $Y = O_{{\mathcal{L}}_{s+j,s}}(1)$ for some $j$ , then for $0\leqslant t \leqslant \tau$ ,

(4.77) \begin{equation} \lVert {(A_t^Y f)(t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \leqslant \int _0^t e^{C(t-s)} \lVert {Y \psi _\circ\, f(r)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}}\, \text{d}r \leqslant e^{C \tau } \lVert {f}\rVert _{B_\tau ^{s+j}}. \end{equation}

Hence, since (c.f. (4.27))

(4.78) \begin{equation} {\mathcal{R}}_1[f](t) = - i \kappa \left (A_t^{{\overline {R}} X X^*} A^X_{(\cdot)}\, f \right )(t) + i \kappa \left (A^{X_2}_t f\right )(t), \end{equation}

estimating this right-hand side gives us the estimate (4.75) for $n=1$ .

The $n=2$ case can also be seen explicitly, using the expansion (4.38). In particular, we have

(4.79) \begin{equation} L_2 = {\overline {R}} X_2 = O_{{\mathcal{L}}_{s+2,s}}(1), \end{equation}

and estimating

(4.80) \begin{align} \lVert {\left (A_t^{{\overline {R}} X X^*} \overline {U}_{(\cdot)}\right )(t) u}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} & \leqslant \int _0^t e^{C(t-r)} \lVert {{\overline {R}} X X^* \overline {U}_r u}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}}\, \text{d}r \nonumber \\ &\lesssim e^{Ct} \lVert {u}\rVert _{H^{s+2, \kappa }_{\mathrm {nucl}}} \end{align}

for some constant $C \gt 0$ , we obtain

(4.81) \begin{equation} \tilde L_2(t) = \overline {U}_t {\overline {R}} X_2 - \left (A_t^{{\overline {R}} X X^*} \overline {U}_{(\cdot)}\right )(t) {\overline {R}} X = O_{{\mathcal{L}}_{s+3,s}}(e^{Ct}). \end{equation}

We can also see that

(4.82) \begin{align} {\mathcal{R}}_2[f](t) &= \left (A_t^{X_3 + {\overline {R}} X X^* {\overline {R}} X} f \right )(t) + \left ( A_t^{{\overline {R}} X X^* } A_{(\cdot)}^{X_2}\, f \right )(t) \nonumber \\ &\quad + \left (A_t^{{\overline {R}} X_2 X^*} A^X_{(\cdot)}\, f \right )(t) + \left ( A_t^{{\overline {R}} X X^*} A_{(\cdot)}^{{\overline {R}} X X^*} A_{(\cdot)}^{X}\, f \right )(t). \end{align}

By Proposition 4.7, $X_3 = O_{{\mathcal{L}}_{s+3,s}}(1)$ . Recall also that for terms of the form $\left ( A^{Y_1}_t \dots A^{Y_n}_{(\cdot)} u \right )(t),$ by their length we mean the number of compositions of $A_t^{Y_j}$ that appear (for this generic term, the length is $n$ ). Then, the total differential order of such a term will be the sum of the differential orders of the $Y_1, \dots , Y_n$ operators.

On the right-hand side of (4.82), we see that the term of length $1$ ,

(4.83) \begin{equation} \left (A_t^{X_3 + {\overline {R}} X X^* {\overline {R}} X} f \right )(t), \end{equation}

is of differential order $3$ . The terms of length $2$ ,

(4.84) \begin{equation} \left ( A_t^{{\overline {R}} X X^* } A_{(\cdot)}^{X_2}\, f \right )(t) + \left (A_t^{{\overline {R}} X_2 X^*} A^X_{(\cdot)}\, f \right )(t), \end{equation}

are of differential order $4$ , and finally the term of length $3$ ,

(4.85) \begin{equation} \left ( A_t^{{\overline {R}} X X^*} A_{(\cdot)}^{{\overline {R}} X X^*} A_{(\cdot)}^{X}\, f \right )(t), \end{equation}

is of differential order $5$ . The estimate (4.75) for $n=2$ follows.

We now proceed to argue by induction, using the previous estimates as the base case. Following the estimate for ${\mathcal{R}}_2[f](t)$ , our intuition is that the length of a term dictates the differential order. Hence, assume that in the expansion of order $n$ , for all $1 \leqslant j \leqslant n+1$ , we have

(4.86) \begin{equation} \sum _{l=1}^{N(n,j)} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} A^{Y_{n, j}^{l,1}}_{(\cdot)}\, f \right )(t) = O_{{\mathcal{L}}_{s+n+j+1}}(e^{Ct}). \end{equation}

Expanding $A^{Y_{n, j}^{l,1}}_{(\cdot)}\, f$ using the expansion (4.19) with $Y = Y_{n, j}^{l,1}$ for all $1 \leqslant l \leqslant N(n,j)$ yields

(4.87) \begin{align} &\sum _{l=1}^{N(n,j)} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} A^{Y_{n, j}^{l,1}}_{(\cdot)}\, f \right )(t) \nonumber \\ &= (\!-\!i\kappa ) \sum _{l=1}^{N(n,j)} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2} {\overline {R}} Y_{n,j}^{l,1}}_{(\cdot)}\, f \right )(t) \end{align}

(4.88) \begin{align} &\quad - (\!-\!i\kappa ) \sum _{l=1}^{N(n,j)} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} \overline {U}_{(\cdot)} \right )(t) {\overline {R}} Y_{n,j}^{l,1} \psi _\circ\, f_0 \end{align}

(4.89) \begin{align} &\quad - (\!-\!i\kappa ) \sum _{l=1}^{N(n,j)} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} A^{W(Y_{n, j}^{l,1})}_{(\cdot)}\, f \right )(t) \end{align}

(4.90) \begin{align} &\quad + (\!-\!i \kappa ) \sum _{l=1}^{N(n,j)} \left ( A^{Y_{n, j}^{l,j}}_t A^{Y_{n, j}^{l,j-1}}_{(\cdot)} \dots A^{Y_{n, j}^{l,2}}_{(\cdot)} A^{{\overline {R}} Y_{n, j}^{l,1} X^*}_{(\cdot)} A^{X}_{(\cdot)}\, f \right )(t).\\[8pt] \nonumber \end{align}

The term (4.87) is of length $j-1$ and is of the same differential order as the term before expansion, $s+n+j+1$ . The term (4.88), by (4.48), contributes to $\tilde L_{n+1}(t)$ . We remark that it is also of differential order $s+n+j+1$ and return to it later. Next, the terms (4.89) are of length $j$ and are of differential order $s+(n+1)+j+1$ , since by Proposition 4.7, $W(Y^{l,1}_{n,j})$ will be of one differential order higher that $Y_{n,j}^{l,1}$ . Lastly, the terms (4.90) are of length $j+1$ and of differential order $s+(n+1)+(j+2)+1$ , since ${\overline {R}} Y_{n,j}^{l,1} X^*$ is of one differential order higher than $Y_{n,j}^{l,1}$ and $X$ is of differential order $1$ . Hence, for all $n \geqslant 2$ and $1 \leqslant j \leqslant n+1$ , (4.86) follows.

Using (4.86), we can now estimate $L_n$ , $\tilde L_n(t)$ and ${\mathcal{R}}_n[f](t)$ using the recursive relations (4.46)–(4.48). We have $Y_{n-1} = O_{{\mathcal{L}}_{s+n,s}}(1)$ , so (4.46) yields the first estimate in (4.74). Next, in (4.48), we see that the term of highest differential order is

(4.91) \begin{equation} \left ( A^{{\overline {R}} X X^*}_t \left (\prod _{1}^{n-2} A^{{\overline {R}} X X^*}_{(\cdot)}\right ) \overline {U}_{(\cdot)} \right )(t) {\overline {R}} X = O_{{\mathcal{L}}_{s+2n-1}}(e^{Ct}), \end{equation}

so the second estimate in (4.74) follows. Finally, in (4.43), see that the term of highest differential order is

(4.92) \begin{equation} \left\lVert {\left ( A^{{\overline {R}} X X^*}_t \left (\prod _{1}^{n-1} A^{{\overline {R}} X X^*}_{(\cdot)}\right ) A^{X}_{(\cdot)}\, f \right )(t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim e^{Ct} \right\lVert {f}\rVert _{B^{s+2n+1}_t}, \end{equation}

and the estimate (4.75) follows.

Now we can estimate the operators (4.51)–(4.54).

Proposition 4.9. Let Assumptions [A1]–[A4] hold. Then, the following estimates hold for $s \geqslant 0$ a positive integer and some constant $C \gt 0$ depending on $s$ and independent of $\kappa$ . For $n \geqslant 1$ and $t \in {\mathbb{R}}$ ,

(4.93) \begin{equation} \lVert {w_n f(t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim \lVert {f(t)}\rVert _{H^{s+n+1, \kappa }_{\mathrm {nucl}}}, \quad \lVert {\tilde w_n(t) f_0}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim e^{Ct} \lVert {f_0}\rVert _{H^{s+2n, \kappa }_{\mathrm {nucl}}}. \end{equation}

For $n \geqslant 1$ and $0 \leqslant t \leqslant \tau$ ,

(4.94) \begin{equation} \lVert {w^\kappa _n[f](t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim e^{C \tau } \left ( \lVert {f_0}\rVert _{H^{s+2n, \kappa }_{\mathrm {nucl}}} + \lVert {f}\rVert _{B_\tau ^{s+2n+2}} \right )\!. \end{equation}

Proof of Proposition 4.9. The estimates follow directly from the definitions (4.51)–(4.54) and the estimates in Proposition 4.8. We illustrate the estimate of $w_n f(t)$ . Using Cauchy-Schwartz and the normalisation $\lVert {\psi _\circ }\rVert _{{\mathcal{H}}_{\mathrm {el}}} = 1$ , it follows from (4.52) that

(4.95) \begin{equation} \lVert {w_n f(t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim \lVert {X^* L_n \psi _\circ\, f}\rVert _{{\mathcal{H}}_{\mathrm {el}} H^{s, \kappa }_{\mathrm {nucl}}}. \end{equation}

Recall that $X^* = \frac {i}{\kappa } P [T, P] {\overline {P}} = O_{{\mathcal{L}}_{s+1,s}}(1)$ (see Lemma C.5 part a)). Combining this with $L_n = O_{{\mathcal{L}}_{s+n,s}}(1)$ (from Proposition 4.8), we obtain the first estimate in (4.93). The remaining estimates follow similarly. In particular, estimating the right-hand side of (4.54) yields

(4.96) \begin{equation} \lVert {w^\kappa _n[f](t)}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} \lesssim \lVert {f(t)}\rVert _{H^{s+n+1, \kappa }_{\mathrm {nucl}}} + e^{C \tau } \left ( \lVert {f_0}\rVert _{H^{s+2n, \kappa }_{\mathrm {nucl}}} + \lVert {f}\rVert _{B_\tau ^{s+2n+2}} \right )\!. \end{equation}

(4.94) follows by using $\lVert {f(t)}\rVert _{H^{s+n+1, \kappa }_{\mathrm {nucl}}} \leqslant \lVert {f}\rVert _{B^{s+n+1}_t} \lesssim \lVert {f}\rVert _{B^{s+2n+2}_t}$ .

The proof of Theorem1.2 (b) now follows immediately by combining the expansion (4.51)–(4.54) with the estimates in Proposition 4.9.

5.1. Main steps of the proof

We first describe the approach and main steps of the proof of Theorem1.3 and outline the proof. The approach is very similar to the proof of Theorem1.1: we seek to re-write the difference $\Psi (t) - (Q_P\tilde {f})(t)$ as the integral in time of some operator-valued quantities, and we show these quantities are small by using the NAIP (the left-handed version, Lemma 2.3).

As in the theorem, let $\Psi (t)$ satisfy the Schrödinger equation (1.14) with $\Psi (0) = \psi _\circ\, f_0$ . Then, using Theorem1.2 part (a) and the definition of $Q_P$ , see (1.29), we can write

(5.1) \begin{equation} \Psi (t) = (Q_P f)(t) = \psi _\circ\, f(t) + \int _0^t \overline {U}_{t-s} X^* \psi _\circ\, f(s)\, \text{d}s, \end{equation}

where recall $X^* = -\frac {i}{\kappa } {\overline {P}} H_\kappa P$ , and $f(t)$ solves the effective nuclear dynamics (1.30), i.e. the dynamics under the full non-local effective nuclear Hamiltonian $h_{\mathrm {eff}}^\kappa$ given in (1.31) with initial condition $f(0) = f_0$ . This representation implies

(5.2) \begin{align} \Psi (t) - (Q_P\tilde {f})(t) &= \psi _\circ \left (f(t) - \tilde {f}(t) \right ) + \int _0^t \overline {U}_{t-s} X^* \psi _\circ \left (f(s) - \tilde {f}(s) \right )\, \text{d}s, \end{align}

where recall $\tilde {f}(t)$ is the solution of the truncated effective nuclear (1.39), under the truncated local effective nuclear Hamiltonian $h_{\mathrm {eff}}^{(2)}$ . In order to proceed, we prove $\psi _\circ \left (f(t) - \tilde {f}(t) \right ) = O_{{\mathcal{L}}_{s+6,s}}(\kappa ^2 e^{C \tau })$ for $0 \leqslant t \leqslant \tau$ by recasting the difference as the integral in time of some operator-valued quantities. We then prove that such integral terms are small using the left-handed NAIP of Lemma 2.3, (2.9). We apply the NAIP with the same choices as before, i.e. $A = \frac {i}{\kappa } \overline {H}$ , $B = \frac {i}{\kappa } \overline {K}$ (where recall $K = E + T$ and $\overline {K} = {\overline {P}} K {\overline {P}}$ ), and the corresponding propagators

(5.3) \begin{align} e^{As} &= e^{i\overline {H} s/\kappa } \;=\!:\; \overline {U}_s, \quad \overline {H} = {\overline {P}} H_\kappa {\overline {P}}, \end{align}

(5.4) \begin{align} e^{Bs} &= e^{i \overline {K} s/\kappa } \;=\!:\; \overline {V}_s, \quad \overline {K} = {\overline {P}} K {\overline {P}},\\[8pt] \nonumber \end{align}

and the reduced resolvent

(5.5) \begin{equation} (A-B)^{-1} = -i \kappa {\overline {P}} (\overline {H}_{\mathrm {bo}} - \overline {E})^{-1}{\overline {P}} \;=\!:\; - i \kappa {\overline {R}}. \end{equation}

As in the previous proofs, we make use of Lemmas 3.1, A.2 and A.3, the commutator estimates of Appendix C, and the propagator estimates of Appendix D.

5.2. Representation formulas

As a first step in the proof of Theorem1.3, we prove the self-adjointness of $h_{\mathrm {eff}}^{(2)}$ in the following lemma.

Lemma 5.1. Let Assumptions [A1]–[A4] hold. a) For $\kappa$ sufficiently small,

(5.6) \begin{equation} 0 \lt \kappa \lt \sqrt {\frac {2}{\tilde C \sqrt {m(m+1)}}}, \quad \tilde C = \tilde C (\delta , \lVert {\nabla _{y_j} H_{\mathrm {bo}}}\rVert , \dots\!), \end{equation}

where $\tilde C$ is the smallest constant such that $\lVert {w_1 u}\rVert \leqslant \tilde C \lVert {u}\rVert _{H^{2,\kappa }_{\mathrm {nucl}}}$ , the operator $h_{\mathrm {eff}}^{(2)}$ is self-adjoint on ${\mathcal{H}}_{\mathrm {nucl}}$ with domain $H^{2, \kappa }_{\mathrm {nucl}}$ . b) The operator $H^{(2)}_\kappa \;:\; {\mathrm {Ran}}\, P \to {\mathrm {Ran}}\, P$ acting via $H^{(2)}_\kappa (\psi _\circ\, f) = \psi _\circ (h_{\mathrm {eff}}^{(2)}\, f )$ is also self-adjoint. It generates the evolution

(5.7) \begin{equation} U^{(2)}_t = e^{- i H^{(2)}_\kappa t/\kappa }, \end{equation}

which for all $\Psi \in {\mathrm {Ran}}\, P$ acts via

(5.8) \begin{equation} U^{(2)}_t \Psi = \psi _\circ e^{- i h_{\mathrm {eff}}^{(2)} t/\kappa } g, \end{equation}

where $g = \langle \psi _\circ , \Psi \rangle _{\mathcal{H}_{\mathrm {el}}}$ .

Proof of Lemma 5.1. Proof of a). Using the Fourier transform on ${\mathbb{R}}^m$ , we can show

(5.9) \begin{equation} \lVert {u}\rVert _{H^{2,\kappa }_{\mathrm {nucl}}} \leqslant \frac {\sqrt {m(m+1)}}{2} \lVert {T u}\rVert _{{\mathcal{H}}_{\mathrm {nucl}}} + \lVert {u}\rVert _{{\mathcal{H}}_{\mathrm {nucl}}}. \end{equation}

Combining this with the estimate $\lVert {w_1 u}\rVert \leqslant \tilde C \lVert {u}\rVert _{H^{2,\kappa }_{\mathrm {nucl}}}$ , implies

(5.10) \begin{equation} \lVert {w_1 u}\rVert _{{\mathcal{H}}_{\mathrm {nucl}}} \leqslant \tilde C \frac {\sqrt {m(m+1)}}{2} \lVert {T u}\rVert _{{\mathcal{H}}_{\mathrm {nucl}}} + \tilde C \lVert {u}\rVert _{{\mathcal{H}}_{\mathrm {nucl}}}, \end{equation}

and hence for $\kappa$ satisfying (5.6), the operator $- \kappa ^2 w_1$ is a relatively bounded perturbation of $T$ with relative bound less than $1$ . It follows easily as well that $w_1$ is a symmetric operator. As $E + \kappa ^2 v$ are bounded symmetric operators, it follows by the Kato-Rellich theorem that $h_{\mathrm {eff}}^{(2)}$ is self-adjoint on ${\mathcal{H}}_{\mathrm {nucl}}$ with domain ${\mathcal{D}}(T) = H^{2,\kappa }_{\mathrm {nucl}}$ .

Proof of b). The self-adjointness of $H^{(2)}_\kappa$ follows immediately. The proof of (5.8) follows analogous to that of Lemma 3.2 b).

Remark. It was shown in Lemma 8 of [Reference Elgart and Hagedorn62] that

(5.11) \begin{equation} \lVert {\nabla _{y_j} P}\rVert \leqslant \frac {1}{\delta } \lVert {\nabla _{y_j} H_{\mathrm {bo}}}\rVert . \end{equation}

Generalising these sorts of estimates would allow us to explicitly compute $\tilde C$ in terms of the spectral gap $\delta$ and the operator norms of derivatives of $H_{\mathrm {bo}}$ (which are all bounded by Assumption [A1]).

We write the difference $\psi _\circ \left (f(t) - \tilde f(t)\right )$ as an operator-valued integral in time in order to integrate by parts.

Lemma 5.2. Let Assumptions [A1]–[A4] hold. Recall $X = \frac {i}{\kappa } {\overline {P}} T P$ and its adjoint $X^* = -\frac {i}{\kappa } P T {\overline {P}}$ . a) We have

(5.12) \begin{equation} \psi _\circ \left (f(t) - \tilde f(t)\right ) = U^{(2)}_{t} Z_t {\overline {R}} X \psi _\circ\, f_0 - \kappa ^2 \int _0^t U^{(2)}_{t-r} \psi _\circ w_2^\kappa [f](r)\, \text{d}r, \end{equation}

where

(5.13) \begin{align} Z_t &= -\kappa ^2 U_{-r}^{(2)} {\overline {R}} X \overline {U}_r \bigg |_{r=0}^{r=t} - \kappa ^2 \int _0^t U^{(2)}_{-r} X_{(2)} \overline {U}_r\, \text{d}r, \end{align}

with

(5.14) \begin{equation} X_{(2)} = \left (X_2\right )^* - i \kappa \psi _\circ w_1 \langle \psi _\circ , X^* \left (\cdot \right ) \rangle _{\mathcal{H}_{\mathrm {el}}}. \end{equation}

Here by $*$ we denote the adjoint and recall $X_2$ is defined in ( 3.27 ).

It is easy to see that the right-hand side of (5.13) is $O(\kappa )$ , so the integration by parts is necessary in order to estimate $Z_t = O(\kappa ^2)$ .

Proof of Lemma 5.2. Using the expansion (1.33) from Theorem1.2 part (b) and introducing $h_{\mathrm {eff}}^{(2)}$ as defined in (1.39), we can write

(5.15) \begin{equation} h_{\mathrm {eff}} f(t) = h_{\mathrm {eff}}^{(2)}\, f(t) + \kappa ^2 \tilde w_1(t) f_0 + i \kappa ^3 w_2^\kappa [f](t). \end{equation}

Using the semigroup $U^{(2)}_t$ and (5.16), we can re-write (1.39) for $\tilde {f}(t)$ and (1.30) for $f(t)$ in Duhamel form, and taking the difference yields

(5.16) \begin{align} \psi _\circ \left (f(t) - \tilde {f}(t) \right ) &= - i \kappa \int _0^t U^{(2)}_{t-r} \psi _\circ \tilde w_1(r) f_0\, \text{d}r \nonumber \\ &\quad - \kappa ^2 \int _0^t U^{(2)}_{t-r} \psi _\circ w_2^\kappa [f](r)\, \text{d}r. \end{align}

By the definition of $\tilde w_1(r)$ , see (4.30), we have

(5.17) \begin{equation} U^{(2)}_{t-r} \psi _\circ \tilde w_1(r) f_0 = U^{(2)}_{t-r} X^* \overline {U}_t {\overline {R}} X \psi _\circ\, f_0, \end{equation}

so we write (5.17) in the form (5.12), with

(5.18) \begin{equation} Z_t = - i \kappa \int _0^t U^{(2)}_{-r} X^* \overline {U}_r\, \text{d}r. \end{equation}

We now integrate $Z_t$ by parts using the left-handed NAIP as described above. We apply the integration by parts expansion (2.9) with $F_r = U^{(2)}_{t-r} X^*$ and $G_r = 1$ (then $R = -i\kappa {\overline {R}}$ ), to the right-hand side of (5.13). In this case $F_r' = \frac {i}{\kappa } U^{(2)}_{t-r} H^{(2)}_\kappa X^*$ , so we obtain

(5.19) \begin{align} Z_t &= - \kappa ^2 U^{(2)}_{t-r} X^* {\overline {R}} \, \overline {U}_r \bigg |_{r=0}^{r=t} + i \kappa \int _0^t U^{(2)}_{t-r} X^* S \overline {U}_r\, \text{d}r \nonumber \\ &\quad - i \kappa \int _0^t U^{(2)}_{t-r} \left ( H^{(2)}_\kappa X^* - X^* \overline {K} \right ) {\overline {R}} \, \overline {U}_{r}\, \text{d}r. \end{align}

Combining the integrals, we obtain (5.14) with

(5.20) \begin{equation} X_{(2)} = -\frac {i}{\kappa } X^* S - \frac {i}{\kappa } \left ( H^{(2)}_\kappa X^* - X^* \overline {K} \right ) {\overline {R}}, \end{equation}

so it remains to show (5.15). Using the definition of $H^{(2)}_\kappa$ and the identity (3.11), we have that for any $g \in {\mathcal{H}}_{\mathrm {nucl}}$ , $H^{(2)}_\kappa \psi _\circ g = K^P \psi _\circ g - \kappa ^2 \psi _\circ w_1 g$ . Since $X^* = P X^*$ , (5.15) follows.

We estimate the right-hand side of (5.12) using (5.14) and (5.15). To do so, we will need the following two additional estimates.

Lemma 5.3. Let Assumptions [A1]–[A4] hold. a) Estimating the right-hand side of ( 5.14 ) yields

(5.21) \begin{equation} Z_t = O_{{\mathcal{L}}_{s+2,s}}(\kappa ^2 e^{Ct}), \end{equation}

for some constant $C \gt 0$ independent of $t$ and $\kappa$ . b) For solutions $f(t)$ of ( 1.30 ) with initial conditions $f(0) = f_0$ , we have

(5.22) \begin{equation} \lVert {f}\rVert _{B^s_\tau } \leqslant C \langle \tau \rangle ^s \lVert {f_0}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}}. \end{equation}

for some constant $C \gt 0$ independent of $\tau$ and $\kappa$ .

Proof of Lemma 5.3. Proof of a). To estimate $X_{(2)}$ , we use the estimate of $X_2 = O_{{\mathcal{L}}_{s+2,s}}(1)$ , which follows from (3.36) in Lemma 3.4, and from the estimate (4.93), we have that $\psi _\circ w_1 \langle \psi _\circ , X^* \left (\cdot \right ) \rangle _{{\mathcal{H}}_{\mathrm {el}}} = O_{{\mathcal{L}}_{s+2,s}}(1)$ . These two estimates together imply

(5.23) \begin{equation} X_{(2)} = O_{{\mathcal{L}}{s+2,s}}(1). \end{equation}

Combining this with the estimates for the propagator $\overline {U}_t = O_{{\mathcal{L}}_{s,s}}(e^{Ct})$ (see TheoremD.1) and using $X = O_{{\mathcal{L}}_{s+1,s}}(1)$ and ${\overline {R}} = O_{{\mathcal{L}}_{s,s}}(1)$ , we obtain (5.21).

Proof of b). From the definition of the operator $Q_P$ given in (1.29), we observe that $\psi _\circ\, f(t) = P (Q_Pf)(t)$ . By Theorem1.2 part (a), $Q_Pf$ is a solution of the full Schrödinger equation $i \kappa \partial _t Q_Pf(t) = H_\kappa Q_Pf(t)$ with initial conditions $(Q_Pf)(0) = \psi _\circ\, f_0$ . Then,

(5.24) \begin{equation} \psi _\circ\, f(t) = P (Q_Pf)(t) = P U_t \psi _\circ\, f_0. \end{equation}

Using this and the fact that $P = O_{{\mathcal{L}}_{s,s}}(1)$ (see Lemma A.2), the definition of the $B^s_\tau$ norm (1.69) implies

(5.25) \begin{align} \lVert {f(t)}\rVert _{B^s_\tau } = \sup _{0 \leqslant t \leqslant \tau } \lVert {\psi _\circ\, f(t)}\rVert _{{\mathcal{H}}_{\mathrm {el}}H^{s, \kappa }_{\mathrm {nucl}}} \lesssim \sup _{0 \leqslant t \leqslant \tau } \lVert {U_t \psi _\circ\, f_0}\rVert _{{\mathcal{H}}_{\mathrm {el}} H^{s, \kappa }_{\mathrm {nucl}}}. \end{align}

The last step is to use the estimate $U_t = O_{{\mathcal{L}}_{s,s}}(\langle t \rangle ^s)$ , which is proven in TheoremD.1, (D.2), on the right-hand side of (5.25). Then the claim follows.

5.3. Conclusion of proof

We now complete the proof of Theorem1.3 using the lemmas we have established.

Proof of Theorem 1.3. Let $\Psi (t)$ satisfy the Schrödinger equation (1.14) with $\Psi (0) = \psi _\circ\, f_0$ and let $\tilde {f}(t)$ be the solution of the truncated effective nuclear (1.39) with $\tilde {f}(0) = f_0$ . We set $\kappa _0$ to the right-hand side of the inequality in (5.6) and consider $\kappa \in (0, \kappa _0)$ . Then, using the definition of $Q_P$ , see (1.29), we can write

(5.26) \begin{align} \Psi (t) - (Q_P\tilde {f})(t) &= \psi _\circ \left (f(t) - \tilde {f}(t) \right ) \nonumber \\ &\quad + \int _0^t \overline {U}_{t-s} X^* \psi _\circ \left (f(s) - \tilde {f}(s) \right )\, \text{d}s, \end{align}

where $f(t)$ solves the effective nuclear dynamics (1.30) with $f(0) = f_0$ . Using the self-adjointness of $h_{\mathrm {eff}}^{(2)}$ (see Lemma 5.1), we can further write

(5.27) \begin{equation} \psi _\circ \left (f(t) - \tilde f(t)\right ) = U^{(2)}_{t} Z_t {\overline {R}} X \psi _\circ\, f_0 - \kappa ^2 \int _0^t U^{(2)}_{t-r} \psi _\circ w_2^\kappa [f](r)\, \text{d}r, \end{equation}

where $Z_t$ , after using the NAIP, admits the expansion (5.14)–(5.15) (see Lemma 5.2).

To estimate the first term on the right-hand side of (5.27), we use Lemma 5.3 part a) to estimate $Z_t$ and the estimate $U^{(2)}_t = O_{{\mathcal{L}}_{s,s}} ( \langle t \rangle ^s)$ , which can be obtained by arguing analogously as in the case of $U^P_t$ (see TheoremD.2). The second term can be estimated by using Lemma 5.3 part b) together with the estimate (1.36) to conclude

(5.28) \begin{equation} \lVert {w^{\kappa }_2 [f](t)}\rVert \lesssim e^{C \tau } \lVert {f_0}\rVert _{H^{s+6, \kappa }_{\mathrm {nucl}}}, \quad 0 \leq t \leq \tau . \end{equation}

Using this on the right-hand side of (5.27), we conclude that

(5.29) \begin{align} \lVert {\psi _\circ \left (f(t) - \tilde {f}(t) \right )}\rVert _{H^{s, \kappa }_{\mathrm {nucl}}} &\lesssim \kappa ^2 \langle t \rangle ^s \lVert {f_0}\rVert _{H^{s+3, \kappa }_{\mathrm {nucl}}} + \kappa ^2 e^{C \tau } \lVert {f_0}\rVert _{H^{s+6, \kappa }_{\mathrm {nucl}}} \nonumber \\ &\lesssim \kappa ^2 e^{C \tau } \lVert {f_0}\rVert _{H^{s+6, \kappa }_{\mathrm {nucl}}}, \quad 0 \leqslant t \leqslant \tau . \end{align}

Using this estimate on the right-hand side of (5.26), we obtain (1.38), and the proof concludes.