This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D706})$. This is a new criterion for determining the multiplicities of eigenvalues for concrete operators.

In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.

This paper deals with the spectral properties of self-adjoint Schrödinger operators with δʹ-type conditions on infinite regular trees. Firstly, we discuss the semi-boundedness and self-adjointness of this kind of Schrödinger operator. Secondly, by using the form approach, we give the necessary and sufficient condition that ensures that the spectra of the self-adjoint Schrödinger operators with δʹ-type conditions are discrete.

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S 2, and investigate the behaviour of the nth eigenvalue λn as a function on such orbits.

For symmetric eigenvalue problems, we constructed a three-term recurrence polynomial filter by means of Chebyshev polynomials. The new filtering technique does not need to solve linear systems and only needs matrix-vector products. It is a memory conserving filtering technique for its three-term recurrence relation. As an application, we use this filtering strategy to the Davidson method and propose the filtered-Davidson method. Through choosing suitable shifts, this method can gain cubic convergence rate locally. Theory and numerical experiments show the efficiency of the new filtering technique.

A self-adjoint first-order system with Hermitian π-periodic potential Q(z), integrable on compact sets, is considered. It is shown that all zeros of are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which Q(z) is π/2-periodic. Furthermore, the zeros of are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which Q(z) = σ 2 Q(z)σ 2. Here, Δ denotes the discriminant of the system and σ 0, σ 2 are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if Q = rσ 0 + qσ 2, for some real-valued π-periodic functions r and q integrable on compact sets.

The Halphen operator is a third-order operator of the form

where g ≠ 2 mod(3), where the Weierstrass ℘-function satisfies the equation

In the equianharmonic case, i.e. g2 = 0, the Halphen operator commutes with some ordinary differential operator Ln of order n ≠ 0 mod(3). In this paper we find the spectral curve of the pair L 3, Ln .

In this paper, we analyze a nonconforming finite element method for the computation of transmission eigenvalues and the corresponding eigenfunctions. The error estimates of the eigenvalue and eigenfunction approximation are given, respectively. Finally, some numerical examples are provided to validate the theoretical results.

We consider the equation

where and

We assume that this equation is correctly solvable in Lp(ℝ). Under these assumptions, we study the problem of compactness of the resolvent of the maximal continuously invertible Sturm–Liouville operator . Here

In the case p = 2, for the compact operator , we obtain two-sided sharp-by-order estimates of the maximal eigenvalue.

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

We show that for self-adjoint Jacobi matrices and Schrödinger operators, perturbed by dissipative potentials in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\ell ^1({\mathbb{N}})$ and $L^1(0,\infty )$ respectively, the finite section method does not omit any points of the spectrum. In the Schrödinger case two different approaches are presented. Many aspects of the proofs can be expected to carry over to higher dimensions, particularly for absolutely continuous spectrum.

In this paper we consider the discreteness of spectrum for higher-order differential operators in weighted function spaces. Using the method of embedding theorems of weighted Sobolev spaces Hnp in weighted spaces Ls,r, we obtain a new sufficient and necessary condition to ensure that the spectrum is discrete, which can be easily used to judge the discreteness of some differential operators.

The current on a linear strip or wire solves an equation governed by a linear integro-differential operator that is the composition of the Helmholtz operator and an integral operator with a logarithmically singular displacement kernel. Investigating the spectral behaviour of this classical operator, we first consider the composition of the second-order differentiation operator and the integral operator with logarithmic displacement kernel. Employing methods of an earlier work by J. B. Reade, in particular the Weyl–Courant minimax principle and properties of the Chebyshev polynomials of the first and second kind, we derive index-dependent bounds for the ordered sequence of eigenvalues of this operator and specify their ranges of validity. Additionally, we derive bounds for the eigenvalues of the integral operator with logarithmic kernel. With slight modification our result extends to kernels that are the sum of the logarithmic displacement kernel and a real displacement kernel whose second derivative is square integrable. Employing this extension, we derive bounds for the eigenvalues of the integro-differential operator of a linear strip with the complex kernel replaced by its real part. Finally, for specific geometry and frequency settings, we present numerical results for the eigenvalues of the considered operators using Ritz's methods with respect to finite bases.

We consider a class of singular Schrödinger operators H that act in L2(0,∞), each of which is constructed from a positive function ϕ on (0,∞). Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of ϕ. For a large class of H that have discrete spectrum, we prove that the eigenvalue asymptotics of H does not depend on rapid oscillations of ϕ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.

We considera discontinuous Sturm–Liouville equation togetherwith two supplementary transmission conditions at the point of discontinuity. We suggest our own approach for finding asymptotic approximation formulas for the eigenvalues of such discontinuous problems.

§1. Introduction. In this paper, we consider the spectral functions ρα(μ) for μ ∈ R associated with a Dirac equation on [0, ∞) given by

together with the initial condition

Given two m-tuples of commuting spectral operators on a Hilbert space, T = (T1,…, Tm) and S = (S1,…, Sm), an extended version of Henrici perturbation theorem is obtained for the joint approximate spectrum of S under perturbation by T. We also derive an extended version of Bauer-Fike theorem for such tuples of operators. The method used involves Clifford algebra techniques introduced by McIntosh and Pryde.

We consider the spectral function

associated with the Sturm-Liouville equation

in situations where q(x) →−∞ as x → ∞ and (1.1) is in the Weyl limit-point case at ∞. As usual, q is real-valued and locally integrable in [0, ∞], and our particular concern is where q(x) has the form

where c (>0) is a parameter, s and p are non-negative on [0, ∞], p(x) → ∞ and p(x) = 0{s(x) } as x → ∞. As the boundary condition at x = 0, we take the Dirichlet condition y(0) = 0 for convenience: we can equally take the Neumann condition y′ (0) = 0 or generally a1y(0) + a2y′ (0) = 0 with real a1 and a2.