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Leonid Parnovski

Publications and source records attributed to Leonid Parnovski.

At least 19 recordsLinked to original sources
arXiv2025

Perturbative diagonalization and spectral gaps of quasiperiodic operators on $\ell^2(\mathbb Z^d)$ with monotone potentials

We obtain a perturbative proof of localization for quasiperiodic operators on $\ell^2(\Z^d)$ with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also proof that the spectra of such operators contain infinitely many gaps.

math.SP↗
arXiv2025

On gaps in the spectra of quasiperiodic Schrödinger operators with discontinuous monotone potentials

We show that, for one-dimensional discrete Schrödinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known result of Gordon and del Rio--Makarov--Simon, combined with a way to consider perturbations whose ranges are not necessarily cyclic. The main application of the results is showing that a class of quasiperiodic operators with sawtooth-like potentials, for which such a version of stable localization is known, has Cantor spectra. We also obtain several results on gap filling under rank one perturbations for some general (not necessarily monotone) classes of quasiperiodic operators with discontinuous potentials.

math.SP↗
arXiv2025

Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators

We consider Schrödinger operators $H=-Δ+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis $[λ_*,+\infty)$. We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on a version of the multi-scale analysis in the momentum space with several new ideas introduced along the way.

math-ph↗
arXiv2025

Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues

In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov-Neumann boundary value problem describing small vertical oscillations of an ideal fluid in a container or in a canal with a uniform cross-section. We prove a two-term asymptotic formula for sloshing eigenvalues. In particular, this confirms a conjecture posed by Fox and Kuttler in 1983. We also obtain similar eigenvalue asymptotics for other related mixed Steklov type problems, and discuss applications to the study of Steklov spectral asymptotics on polygons.

math.SP↗
arXiv2023

Classical Wave methods and modern gauge transforms: Spectral Asymptotics in the one dimensional case

In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let $H: L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ H:=-\frac{d^2}{dx^2}+V, $$ where $V$ is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, $\mathbb{1}_{(-\infty,ρ^2]}(H)$, has a complete asymptotic expansion in powers of $ρ$. This settles the 1-dimensional case of a conjecture made by the last two authors.

math.SP↗
arXiv2022

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunctions. Our proofs are based on an explicit construction of quasimodes. We use a variety of methods, including ideas from spectral geometry, layer potential analysis, and some new techniques tailored to our problem.

math.SP↗
arXiv2022

Convergence of perturbation series for unbounded monotone quasiperiodic operators

We consider a class of unbounded quasiperiodic Schrödinger-type operators on $\ell^2(\mathbb Z^d)$ with monotone potentials (akin to the Maryland model) and show that the Rayleigh--Schrödinger perturbation series for these operators converges in the regime of small kinetic energies, uniformly in the spectrum. As a consequence, we obtain a new proof of Anderson localization in a more general than before class of such operators, with explicit convergent series expansions for eigenvalues and eigenvectors. This result can be restricted to an energy window if the potential is only locally monotone and one-to-one. A modification of this approach also allows the potential to be non-strictly monotone and have a flat segment, under additional restrictions on the frequency.

math.SP↗
arXiv2021

The almost periodic gauge transform -- An abstract scheme with applications to Dirac Operators

One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schrödinger operators is the method of gauge transform. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe--Sommerfeld property, that the spectrum contains a semi-axis -- or indeed two semi-axes in the case of operators that are not semi-bounded.

math-ph↗
arXiv2021

Perturbative diagonalisation for Maryland-type quasiperiodic operators with flat pieces

We consider quasiperiodic operators on $\mathbb Z^d$ with unbounded monotone sampling functions ("Maryland-type"), which are not required to be strictly monotone and are allowed to have flat segments. Under several geometric conditions on the frequencies, lengths of the segments, and their positions, we show that these operators enjoy Anderson localization at large disorder.

math.SP↗
arXiv2020

Inverse Steklov spectral problem for curvilinear polygons

This paper studies the inverse Steklov spectral problem for curvilinear polygons. For generic curvilinear polygons with angles less than $π$, we prove that the asymptotics of Steklov eigenvalues obtained in arXiv:1908.06455 determines, in a constructive manner, the number of vertices and the properly ordered sequence of side lengths, as well as the angles up to a certain equivalence relation. We also present counterexamples to this statement if the generic assumptions fail. In particular, we show that there exist non-isometric triangles with asymptotically close Steklov spectra. Among other techniques, we use a version of the Hadamard--Weierstrass factorisation theorem, allowing us to reconstruct a trigonometric function from the asymptotics of its roots.

math.SP↗
arXiv2018

Perturbation theory for almost-periodic potentials I. One-dimensional case

We consider the family of operators $H^{(ε)}:=-\frac{d^2}{dx^2}+εV$ in ${\mathbb R}$ with almost-periodic potential $V$. We study the behaviour of the integrated density of states (IDS) $N(H^{(ε)};λ)$ when $ε\to 0$ and $λ$ is a fixed energy. When $V$ is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each $λ$ the IDS has a complete asymptotic expansion in powers of $ε$; these powers are either integer, or in some special cases half-integer. These results are new even for periodic $V$. We also prove that when the potential is neither periodic nor quasi-periodic, there is an exceptional set $\mathcal S$ of energies (which we call $\hbox{the super-resonance set}$) such that for any $\sqrtλ\not\in\mathcal S$ there is a complete power asymptotic expansion of IDS, and when $\sqrtλ\in\mathcal S$, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set $\mathcal S$ is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of $H^{(ε)}$ has a complete asymptotic expansion in natural powers of $ε$ when $ε\to 0$.

math-ph↗
arXiv2017

Perturbation theory for spectral gap edges of 2D periodic Schrödinger operators

We consider a two-dimensional periodic Schrödinger operator $H=-Δ+W$ with $Γ$ being the lattice of periods. We investigate the structure of the edges of open gaps in the spectrum of $H$. We show that under arbitrary small perturbation $V$ periodic with respect to $NΓ$ where $N=N(W)$ is some integer, all edges of the gaps in the spectrum of $H+V$ which are perturbation of the gaps of $H$ become non-degenerate, i.e. are attained at finitely many points by one band function only and have non-degenerate quadratic minimum/maximum. We also discuss this problem in the discrete setting and show that changing the lattice of periods may indeed be unavoidable to achieve the non-degeneracy.

math-ph↗
arXiv2016

A generalised Gauss circle problem and integrated density of states

Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer lattice of affine planes inside a ball. The first term is the volume of the ball; we study the size of the remainder term. While the classical problem is equivalent to counting eigenvalues of the Laplace operator on the torus, our variation corresponds to the integrated density of states of the Laplace operator on the product of a torus with Euclidean space. The asymptotics we obtain are then used to compute the density of states of the magnetic Schroedinger operator.

math.SP↗
arXiv2014

Asymptotic eigenvalue estimates for a Robin problem with a large parameter

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.

math-ph↗
arXiv2013

The Steklov spectrum of surfaces: asymptotics and invariants

We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neumann operator and on a number-theoretic argument.

math.SP↗
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↗