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Oleg Safronov

Publications and source records attributed to Oleg Safronov.

15 recordsLinked to original sources
arXiv2026

On the Emergence of Discrete Spectrum for Weakly Disordered Schrödinger Operators

We investigate the spectral properties of the Anderson operator perturbed by a localized negative potential, \(-V\). Specifically, we analyze the random Schrödinger operator defined by \(H = -Δ+\ve \sum_{n} ω_n χ_n - V\), where the unperturbed operator exhibits a disordered energy landscape. Our primary focus is to establish precise estimates on the number of negative eigenvalues (bound states) induced by the attractive perturbation. By analyzing the competition between Anderson localization and the binding capacity of the potential, we provide quantitative bounds on the discrete spectrum. These results offer new insights into how randomness enhances the eigenvalue bounds.

math-ph↗
arXiv2025

Spectral theory of Schrödinger operators with potentials that are measures supported on ${\Bbb N}$

We discuss spectral properties of the one-dimensional Schrödinger operator with a potential of the form $\sum V(n)δ(x-n)$. Our main result says that the absolutely continuous spectum of such an operator covers an interval $[α^2,β^2]$, if $V\in \ell^4$ and the Fourier series $\sum e^{2i kn}V(n)$ is a function of $k$ that is square integrable over $[α,β]$. We prove that this result is sharp by constructing examples of potentials $V\notin\ell^2$ for which the spectrum of the Schrödinger operator is singular.

math-ph↗
arXiv2025

Discrete Spectrum of the Bilayer Graphene Operator

We consider the graphene operator $D_m$ perturbed by a decaying potential $αV$, where $α$ is a coupling constant. We study the number $N(λ,α)$ of eigenvalues of the operator $D(t)=D_m-tV$ passing through a regular point $λ\inρ(D_m)$ as $t$ changes from $0$ to $α$. We obtain asymptotic formulas for $N(λ,α)$ as $α\to\infty$.

math.SP↗
arXiv2025

Lyapunov exponent for quantum graphs that are elements of a subshift of finite type

We consider the Schrödinger operator on the quantum graph whose edges connect the points of ${\Bbb Z}$. The numbers of the edges connecting two consecutive points $n$ and $n+1$ are read along the orbits of a shift of finite type. We prove that the Lyapunov exponent is potitive for energies $E$ that do not belong to a discrete subset of $[0,\infty)$. The number of points $E$ of this subset in $[(π(j-1))^2, (πj)^2]$ is the same for all $j\in {\Bbb N}$.

math-ph↗
arXiv2024

Monotonicity of the set of zeros of the Lyapunov exponent with respect to shift embeddings

We consider the discrete Schrödinger operators with potentials whose values are read along the orbits of a shift of finite type. We study a certain subset of the collection of energies at which the Lyapunov exponent is zero and prove monotonicity of this set with respect to the shift embeddings. Then we introduce a certain function ${\mathcal J}(A,μ)$ determined by the position of these zeros and prove monotonicity of ${\mathcal J}(A,μ)$ with respect to embeddings.

math-ph↗
arXiv2018

Eigenvalue bounds for Stark operators with complex potentials

We consider the 3-dimensional Stark operator perturbed by a complex-valued potential. We obtain an estimate for the number of eigenvalues of this operator as well as for the sum of imaginary parts of eigenvalues situated in the upper half-plane.

math.SP↗
arXiv2010

Lower bounds on the eigenvalue sums of the Schrödinger operator and the spectral conservation law

In the first part of the paper we consider the Schrödinger operator $ -Δ-V(x),\quad V>0. $ We discuss the relation between the behavior of $V$ at the infinity and the properties of the negative spectrum of $H$. After that, we consider the case when $V$ changes its sign: $ V=V_+-V_-$, $2V_\pm=|V|\pm V. $ In this case, we treat $V$ and $-V$ symmetrically and study the relation between the behavior of $V$ at the infinity and the negative spectra of the operators $H_+=-Δ+V$ and $H_-=-Δ-V$.

math.SP↗
arXiv2010

On a sum rule for Schrödinger operators with complex potentials

We study the distribution of eigenvalues of the one-dimensional Schrödinger operator with a complex valued potential $V$. We prove that if $|V|$ decays faster than the Coulomb potential, then the series of imaginary parts of square roots of eigenvalues is convergent.

math-ph↗