We consider in L 2 (R d), d ≥ 2, the perturbed polyharmonic operator H = (−) 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… (More)
We consider a two-dimensional innnitely long acoustic waveguide formed by two parallel lines containing an arbitrarily shaped obstacle. The existence of trapped modes that are the eigenfunctions of the Laplace operator in the corresponding domain subject to Neumann boundary conditions was proved by Evans, Levitin & Vassiliev (1994) for obstacles symmetric… (More)
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 eigen-values of the… (More)
We consider a quantity κ(Ω) — the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximized, among all convex balanced domains Ω ⊂ R d of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We… (More)
We prove the complete asymptotic expansion of the integrated density of states of a Schrödinger operator H = −∆ + b acting in R d when the potential b is either smooth periodic, or generic quasi-periodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions.
We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided… (More)
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.
We prove the complete asymptotic expansion of the integrated density of states of a two-dimensional Schrödinger operator with a smooth periodic potential.
We consider a periodic self-adjoint pseudo-differential operator H = (−∆) m + B, m > 0, in R d which satisfies the following conditions: (i) the symbol of B is smooth in x, and (ii) the perturbation B has order less than 2m. Under these assumptions, we prove that the spectrum of H contains a half-line. This, in particular implies the Bethe-Sommerfeld… (More)