Michael Levitin

EXISTENCE THEOREMS FOR TRAPPED MODES

A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.

Commutators, spectral trace identities, and universal estimates for eigenvalues

Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of non-self-adjoint operators.

How large can the first eigenvalue be on a surface of genus two

AbstractSharp upper bounds for the first eigenvalue of the Laplacian on asurface of a fixed area are known only in genera zero and one. Weinvestigate the genus two case and conjecture that the first eigenvalue ismaximized on a singular surface which is realized as a double branchedcovering over a sphere. The six ramification points are chosen in sucha way that this surface has a complex structure of the Bolza surface.We prove that our conjecture follows from a lower bound on the firsteigenvalue of a certain mixed Dirichlet-Neumann boundary value problemon a half-disk. The latter can be studied numerically, and we presentconclusive evidence supporting the conjecture. Keywords: Laplacian, first eigenvalue, surface of genus two, mixed boundaryvalue problem. ∗ Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West,Montreal, QC H3A 2K6, Canada; e-mail jakobson@math.mcgill.ca † Department of Mathematics, Heriot-Watt University, United Kingdom; e-mailM.Levitin@ma.hw.ac.uk

On approximation of the eigenvalues of perturbed periodic Schrödinger operators

This paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic Schrodinger operator perturbed by a fast decreasing potential. We use a recently developed technique, the so-called quadratic projection method, in order to achieve convergence free from spectral pollution. We describe the theoretical foundations of the method in detail and illustrate its effectiveness by several examples.

Isospectral domains with mixed boundary conditions

We construct a series of examples of planar isospectral domains with mixed Dirichlet-Neumann boundary conditions. This is a modification of a classical problem proposed by M Kac.

Existence of eigenvalues of a linear operator pencil in a curved waveguide - localized shelf waves on a curved coast

The question of the existence of nonpropagating, trapped continental shelf waves (CSWs) along curved coasts reduces mathematically to a spectral problem for a self-adjoint operator pencil in a curved strip. Using methods developed for the waveguide trapped mode problem, we show that such CSWs exist for a wide class of coast curvature and depth profiles.

Range of the First Three Eigenvalues of the Planar Dirichlet Laplacian

Extensive numerical experiments have been conducted by the authors, aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M. Ashbaugh and R. Benguria. Some properties of a maximizer of the ratio λ 3 /λ 1 are also proved in the paper.

Eigenvalues of a one-dimensional Dirac operator pencil

We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a non-trivial square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.

Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

A simple method of calculating eigenvalues and resonances in domains with infinite regular ends

A new approach is presented for the solution of spectral problems on infinite domains with regular ends, which avoids the need to solve boundary-value problems for many trial values of the spectral parameter. We present numerical results both for eigenvalues and for resonances, comparing with results reported by Aslanyan, Parnovski and Vassiliev.