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Roman Shterenberg

Publications and source records attributed to Roman Shterenberg.

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

Measuring Heterogeneity in Machine Learning with Distributed Energy Distance

In distributed and federated learning, heterogeneity across data sources remains a major obstacle to effective model aggregation and convergence. We focus on feature heterogeneity and introduce energy distance as a sensitive measure for quantifying distributional discrepancies. While we show that energy distance is robust for detecting data distribution shifts, its direct use in large-scale systems can be prohibitively expensive. To address this, we develop Taylor approximations that preserve key theoretical quantitative properties while reducing computational overhead. Through simulation studies, we show how accurately capturing feature discrepancies boosts convergence in distributed learning. Finally, we propose a novel application of energy distance to assign penalty weights for aligning predictions across heterogeneous nodes, ultimately enhancing coordination in federated and distributed settings.

stat.ML↗
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

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↗
arXiv2022

Solutions of Gross-Pitaevskii Equation with Periodic Potential in Dimension Three

Quasi-periodic solutions of the Gross-Pitaevskii equation with a periodic potential in dimension three are studied. It is proven that there is an extensive "non-resonant" set ${\mathcal G} \subset \mathbb{R}^3$ such that for every $\vec k\in \mathcal G$ there is a solution asymptotically close to a plane wave $Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$ as $|\vec k|\to \infty $, given $A$ is sufficiently small.

math-ph↗
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↗
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↗
arXiv2018

Solutions of Gross-Pitaevskii Equation with Periodic Potential in Dimension Two

Quasi-periodic solutions of a nonlinear polyharmonic equation for the case $4l>n+1$ in $\R^n$, $n>1$, are studied. This includes Gross-Pitaevskii equation in dimension two ($l=1,n=2$). It is proven that there is an extensive "non-resonant" set ${\mathcal G}\subset \R^n$ such that for every $\vec k\in \mathcal G$ there is a solution asymptotically close to a plane wave $Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$ as $|\vec k|\to \infty $, given $A$ is sufficiently small.

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

Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-periodic Potential in Dimension Two

We prove the existence of ballistic transport for the Schrödinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior work to establish the existence of an absolutely continuous component and other spectral properties. The latter include detailed information on the structure of generalized eigenvalues and eigenfunctions. These allow to establish the crucial ballistic lower bound through integration by parts on an appropriate extension of a Cantor set in momentum space, as well as through stationary phase arguments.

math-ph↗
arXiv2014

Extended States for the Schrödinger Operator with Quasi-periodic Potential in Dimension Two

We consider a Schrödinger operator $H=-Δ+V(\vec x)$ in dimension two with a quasi-periodic potential $V(\vec x)$. We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i\langle \vec \varkappa,\vec x\rangle}$ at the high energy region. Second, the isoenergetic curves in the space of momenta $\vec \varkappa$ corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results. The result is based on the previous paper [1] on quasiperiodic polyharmonic operator $(-Δ)^l+V(\vec x)$, $l>1$. We address here technical complications arising in the case $l=1$. However, this text is self-contained and can be read without familiarity with [1].

math-ph↗
arXiv2014

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,Ω}$, the Krein--von Neumann extension of the perturbed Laplacian $-Δ+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(Ω)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Ω\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.

math.SP↗
arXiv2013

Stability for the inverse resonance problem for the CMV operator

For the class of unitary CMV operators with super-exponentially decaying Verblunsky coefficients we give a new proof of the inverse resonance problem of reconstructing the operator from its resonances - the zeros of the Jost function. We establish a stability result for the inverse resonance problem that shows continuous dependence of the operator coefficients on the location of the resonances.

math.SP↗