Related papers: Remarks on interior transmission eigenvalues, Weyl…
The distribution of eigenvalues of the wave equation in a bounded domain is known as Weyl's problem. We describe several computational projects related to the cumulative state number, defined as the number of states having wavenumber up to…
This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random…
We derive combinatorial identities for variables satisfying specific systems of commutation relations, in particular elliptic commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and…
The purpose of this paper is to review the asymptotic distribution of eigenvalues of the Dirichlet Laplacian. We introduce and recall all the relevant spectral quantities and provide a proof based on the Fourier Tauberian Theorem.
We study the counting function of the eigenvalues for tensor products of operators, and their perturbations, in the context of Shubin classes and closed manifolds. We emphasize connections with problems of analytic number theory, concerning…
Let $M$ be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group $G$. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on…
The billiard problem concerns a point particle moving freely in a region of the horizontal plane bounded by a closed curve $\Gamma$, and reflected at each impact with $\Gamma$. The region is called a `billiard', and the reflections are…
We derive semiclassical contributions of periodic orbits from a boundary integral equation for three-dimensional billiard systems. We use an iterative method that keeps track of the composition of the stability matrix and the Maslov index…
A boundary integral equation formulation is presented for the electromagnetic transmission problem where an incident electromagnetic wave is scattered from a bounded dielectric object. The formulation provides unique solutions for all…
Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. A well known classical constant of the motion has in the quantum case an ill-defined commutator with the Hamiltonian. It is shown how this problem…
We investigate the transmission and reflection survival probabilities for the chaotic stadium billiard with two holes placed asymmetrically. Classically, these distributions are shown to have algebraic or exponential decays depending on the…
We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away…
We study periodic infinite billiards in the plane. We show that for rational models, some particular obstacles can be added periodically, so that the billiard flow in the resulting table is recurrent in almost every direction.
We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed G-manifold M, where G is a compact, connected Lie group acting effectively and isometrically on M. Using resolution of singularities, we…
We study an eigenvalue problem in the framework of double phase variational integrals and we introduce a sequence of nonlinear eigenvalues by a minimax procedure. We establish a continuity result for the nonlinear eigenvalues with respect…
In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in H\"older spaces. As an…
We present numerical results on calculations of energy spectra and wave functions of annular billiard.
We propose geometric tools that are suitable for studying the behavior of a billiard trajectory in a homogeneous force field. Two examples are considered: a vertical plane with an open top and with a parabolic or right angle boundary at the…
We survey several generalizations of the Weyl algebra including generalized Weyl algebras, twisted generalized Weyl algebras, quantized Weyl algebras, and Bell-Rogalski algebras. Attention is paid to ring-theoretic properties,…
We present analytical and numerical solutions of the Lippmann-Schwinger equation for the scattered wavefunctions generated by confocal parabolic billiards and parabolic segments with various $\delta$-type potential-strength functions. The…