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It is a general property of elliptic differential operators with periodic coefficients, that their spectra are formed by union of closed intervals called spectral bands (see [12], [14]) possibly separated by gaps. One of the challenging questions of the spectral theory of periodic operators is to find out whether or not the number of gaps in the spectrum of… (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.

- Leonid Parnovski
- 2009

Abstract. We consider a periodic self-adjoint pseudo-differential operatorH = (−∆)m+ B, m > 0, in R 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)

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)

- Leonid Parnovski
- 2010

We prove the complete asymptotic expansion of the integrated density of states of a Schrödinger operator H = −∆+b acting in R 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.

- Leonid Parnovski
- 2007

- R. Hill, Leonid Parnovski
- 2005

The problem of estimating the number of points of a lattice that lie in a ball, is often called the circle problem. In the case of lattices in Euclidean space, this question goes back at least as far as Gauss. If we call Nρ the number of points of Z inside the ball B(0, ρ), then one easily sees that the leading term of Nρ is the area, πρ, of B(0, ρ). It is… (More)

- Leonid Parnovski
- 2007

We consider Schrödinger operator −∆+V in R (d ≥ 2) with smooth periodic potential V and prove that there are only finitely many gaps in its spectrum. Dedicated to the memory of B.M.Levitan

- Leonid Parnovski
- 2008

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

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)