Leonid Parnovski

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.

On the Bethe-Sommerfeld conjecture for the polyharmonic operator

We consider in

Asymptotic eigenvalue estimates for a Robin problem with a large parameter

L2( R d), d≥2, the perturbed polyharmonic operator H=(−Δ)l+V, l>0, with a function V periodic with respect to a lattice in R d. We prove that the number of gaps in the spectrum of H is finite if 6l>d+2. Previously the finiteness of the number of gaps was known for 4l>d+1. The proof is based on arithmetic properties of the lattice and elementary perturbation theory.

Isospectral domains with mixed boundary conditions

Robin problem for the Laplacian in a bounded planar domain with a smooth boundary and a large parameter in the boundary condition is considered. We prove a two-sided three-term asymptotic estimate for the negative eigenvalues. Furthermore, improving the upper bound we get a two term asymptotics in terms of the coupling constant and the maximum of the boundary curvature.

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

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.

Spectral asymptotics of Laplace operators on surfaces with cusps

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.

The Steklov spectrum of surfaces: Asymptotics and invariants

Let M be a connected surface with cusps (i.e., M is a compact perturbation of a surface with constant negative curvature and finite volume). Let A be the selfadjoint extension of the positive Laplace operator on M. Then (see [8] and references there) the spectrum of A consists of: (i) the finite number of eigenvalues 0 = 2o < 21 =< ... < 2t < 1/4, (ii) the absolutely continuous spectrum [1/4, +oo) (iii) the eventual eigenvalues 1/4 =< 2r+1 _-< ... (in finite number or not) which are embedded in the continuous spectrum. Let Na(T) and Nc(T) be the counting functions of the discrete and continuous spectra correspondingly (see Sect. 2 for the precise definition). Because of the complicated structure of spectrum (existence of embedded eigenvalues), it is hard to compute the asymptotics of Nd(T) or No(T) separately. However, it is possible to study the asymptotics of the sum

Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator

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Bethe–Sommerfeld Conjecture for Pseudodifferential Perturbation

We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrödinger operator with a smooth periodic potential.

Inverse electrostatic and elasticity problems for checkered distributions

We consider a periodic pseudodifferential operator H = (− Δ) l + A (l > 0) in R d which satisfies the following conditions: (i) the symbol of H is smooth in x, and (ii) the perturbation A has order smaller than 2l − 1. Under these assumptions, we prove that the spectrum of H contains a half-line.