We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.

We consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form ${\pi}/{2q}$, where q is an integer. We are interested in finding a two-term asymptotic expansion of the eigenvalue counting function. When both angles are ${\pi}/{4}$, we compute the exact value of the second term. As for the general case, we conjecture an asymptotic expansion by constructing quasimodes for the problem and computing the counting function of the related quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem and correspond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvalues are exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion of their counting function is closely related to a lattice counting problem inside a perturbed ellipse where the perturbation is in a sense random. The contribution of the angles can then be detected through that perturbation.

In this paper we study the small-scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is, the coefficients take value $+1$ with probability p and $-1$ with probability $1-p$. Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ($p=0.5$) coin to retain equidistribution.

For a family of elliptic operators with periodically oscillating coefficients, $-{\rm div}(A(\cdot /\varepsilon )\nabla )$ with tiny ε > 0, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions (eigenspaces) for both the Dirichlet and Neumann problems in bounded, smooth and strictly convex domains (or more general domains of finite type). A new first-order correction term is introduced to derive the expansion of eigenfunctions in L2 or $H^1_{\rm loc}$. Our results rely on the recent progress on the homogenization of boundary layer problems.

This paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux, we establish a Hardy-type inequality. In the regime with an infinite discrete spectrum, we obtain sharp spectral asymptotics with a refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations.

We study the fine-scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension $d\geqslant 3$. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the $(d-2)$-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids.

This paper is devoted to dimensional reductions via the norm-resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different partial differential equation problems from various areas of mathematical physics; all are analysed in a unified manner, known results are recovered and new ones established.

We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$ , where $P$ is an operator of order 0 with geometric origin and $f$ a multiplication operator by a function. When $f$ is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions $f$ , displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

We consider an elliptic self-adjoint first-order differential operator $L$ acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of the operator $L$ is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Given a positive scalar weight function, we study the weighted eigenvalue problem for the operator $L$ . The corresponding counting function (number of eigenvalues between zero and a positive $\unicode[STIX]{x1D706}$ ) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as $\unicode[STIX]{x1D706}\rightarrow +\infty$ and we have recently derived an explicit formula for the second asymptotic coefficient. The purpose of this paper is to establish the geometric meaning of the second asymptotic coefficient. To this end, we identify the geometric objects encoded within our eigenvalue problem—metric, non-vanishing spinor field and topological charge—and express our asymptotic coefficients in terms of these geometric objects. We prove that the second asymptotic coefficient of the counting function has the geometric meaning of the massless Dirac action.

We prove sufficient and necessary conditions for compactness of the Sobolev embeddings of Besov and Triebel–Lizorkin spaces defined on bounded and unbounded uniformly E-porous domains. The asymptotic behaviour of the corresponding entropy numbers is calculated. Some applications to the spectral properties of elliptic operators are described.

We prove a sharp upper bound on the L2 norm of a GL3 Maass form restricted to GL2×ℝ+.

We study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.

In this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.