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Leonid Parnovski

Publications and source records attributed to Leonid Parnovski.

29 records · Page 2Linked to original sources
arXiv2012

On a class of inverse electrostatic and elasticity problems

We study the inverse electrostatic and elasticity problems associated with Poisson and Navier equations. The uniqueness of solutions of these problems is proved for piecewise constant electric charge and internal stress distributions having a checkered structure: they are constant on rectangular blocks. Such distributions appear naturally in practical applications. We also discuss computational challenges arising in the numerical implementation of our method.

math.AP↗
arXiv2012

Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic pseudo-differential operators

We obtain a complete asymptotic expansion of the integrated density of states of operators of the form H =(-Δ)^w +B in R^d. Here w >0, and B belongs to a wide class of almost-periodic self-adjoint pseudo-differential operators of order less than 2w. In particular, we obtain such an expansion for magnetic Schrödinger operators with either smooth periodic or generic almost-periodic coefficients.

math-ph↗
arXiv2009

Fourier transform, null variety, and Laplacian's eigenvalues

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 $Ω\subset\Rbb^d$ 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 prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between $κ(Ω)$ and the eigenvalues of the Laplacians.

math.SP↗
arXiv2008

Bethe-Sommerfeld Conjecture

We consider Schroedinger operator $-Δ+V$ in $R^d$ ($d\ge 2$) with smooth periodic potential $V$ and prove that there are only finitely many gaps in its spectrum.

math.SP↗
arXiv2005

On the principal eigenvalue of a Robin problem with a large parameter

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 estimates for this term in a variety of situations.

math.SP↗
arXiv2004

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

The study of the possibility of existence of the non-propagating, 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 the methods developed in the setting of the waveguide trapped mode problem, we show that such CSWs exist for a wide class of coast curvature and depth profiles.

math.SP↗
arXiv2001

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 (Payne-Polya-Weinberger, Hile-Protter, etc.), 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.

math.SP↗