Leonid Parnovski

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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 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)
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)