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Grigori Rozenblum

Publications and source records attributed to Grigori Rozenblum.

At least 19 recordsLinked to original sources
arXiv2026

Negative eigenvalue estimates for polyharmonic Schrödinger operators with measure-potentials: the subcritical case

We study spectral estimates for polyharmonic Schrödinger operators $-Δ^l-μ$ in the subcritical regime $2l<\mathbf{N}$. The measure potential $μ$ is assumed to satisfy a capacitary smallness condition which guarantees that the corresponding operator is semibounded and self-adjoint. With such a measure $μ$ we associate an Otelbaev function, which reflects both the local concentration and the spatial distribution of the potential. In terms of this function, we obtain two-sided estimates for the distribution function of the negative eigenvalues, and derive a sufficient condition and a necessary condition for the discreteness of the negative spectrum. As an application, we establish two-sided estimates of Lieb-Thirring-type, improving the classical ones.

math.SP↗
arXiv2026

Higher order H{ö}lder approximation by solutions of second order elliptic equations

For a given second order elliptic operation $\mathcal{L}$ in a domain $Ω\subset{\mathbb{R}}^\mathbf{N}$, $\mathbf{N}\ $, and a compact set $\mathbf{K}\subsetΩ$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space $\mathcal{H}^{\mathbf{r}+ω}_{\mathcal{L}}(\mathbf{K})$ of continuous functions $f(x),\, x\in\mathbf{K}$, admitting, for any $δ>0$, a local approximation in the $δ$-neighborhood of any point $x\in\mathbf{K}$, with $δ^{\mathbf{r}}ω(δ)$-error estimate, by solutions of the equation $\mathcal{L} u=0$. For such functions, we prove the existence of a global approximation $v_δ$ on $\mathbf{K}$ with the same order of error estimate, by a solution of the same equation in a $δ$-neighborhood of $\mathbf{K}$. A number of properties of these functions $v_δ$ and their derivatives are established.

math.AP↗
arXiv2025

Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures

We consider operators of the form $\mathbf{T}=\mathbf{A^*}(Vμ)\mathbf{A}$ in $\mathbb{R}^\mathbf{N}$, where $\mathbf{A}$ is a pseudodifferential operator of order $-l$, $μ$ is a compactly supported singular measure, order $s>0$ Ahlfors-regular, and $V$ is a weight function on the support of $μ$. The scalar type operator $\mathbf{A}$ and the weight function $V$ are supposed to be $m\times m$ matrix valued. We establish Weyl type asymptotic formulas for singular numbers and eigenvalues of $\mathbf{T}$ for $μ$ being the natural measure on a compact Lipschitz surface. For a general Ahlfors-regular measure $μ$, we prove that the previously found upper spectral estimates are order sharp.

math.SP↗
arXiv2024

Spectral properties of the resolvent difference for singularly perturbed operators

We obtain order sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators $\mathbf{A}+\mathbf{V}_1$ and $\mathbf{A}+\mathbf{V}_2$ in a domain $Ω\subseteq \mathbb{R}^\mathbf{N}$ with perturbations $\mathbf{V}_1, \mathbf{V}_2$ generated by $V_1μ,V_2μ,$ where $μ$ is a measure singular with respect to the Lebesgue measure and satisfying two-sided or one-sided conditions of Ahlfors type, while $V_1,V_2$ are weight functions subject to some integral conditions. As an important special case, spectral estimates for the difference of resolvents of two Robin realizations of the operator $\mathbf{A}$ with different weight functions are obtained. For the case when the support of the measure is a compact Lipschitz hypersurface in $Ω$ or, more generally, a rectifiable set of Haußdorff dimension $d=\mathbf{N}-1$, the Weyl type asymptotics for eigenvalues is justified.

math.SP↗
arXiv2024

Negative eigenvalue estimates for the 1D Schr{ö}dinger operator with measure-potential

We investigate the negative part of the spectrum of the operator $-\partial^2 - μ$ on $L^2(\mathbb R)$, where a locally finite Radon measure $μ\geq 0$ is serving as a potential. We obtain estimates for the eigenvalue counting function, for individual eigenvalues and estimates of the Lieb-Thirring type. A crucial tool for our estimates is Otelbaev's function, a certain average of the measure potential $μ$, which is used both in the proofs and the formulation of most of the results.

math.SP↗
arXiv2022

Topological obstructions to the diagonalisation of pseudodifferential systems

Given a matrix pseudodifferential operator on a smooth manifold, one may be interested in diagonalising it by choosing eigenvectors of its principal symbol in a smooth manner. We show that diagonalisation is not always possible, on the whole cotangent bundle or even in a single fibre. We identify global and local topological obstructions to diagonalisation and examine physically meaningful examples demonstrating that all possible scenarios can occur.

math.AP↗
arXiv2022

Aymptotics of eigenvalues of the Neumann-Poincar'e operator in 3D elasticity

We consider the Neumann-Poincar'e (double layer potential) operator in 3D elasicity on a smooth closed surface. Its essential spectrum consists of 3 points. We find the asymptotics of sequences of eigenvalues converging to these three pounts. They are expressed via the eigenvalue distribution for a matrix eigenvalue problem depending on the points of the cotangent bundle of the surface. For the two-sided asymptotics for the distribution of the the union of sequences of eigenvalues converging to the point of the essential spectrum from both sides, an expression is found via the Euler characteristics and the Willmore energy of the surface.

math.AP↗
arXiv2022

Spectral estimates and asymptotics for integral operators on singular sets

For singular numbers of integral operators of the form $u(x)\mapsto \int F_1(X)K(X,Y,X-Y)F_2(Y)u(Y)μ(dY),$ with measure $μ$ singular with respect to the Lebesgue measure in $\mathbb{R}^\mathbf{N}$, order sharp estimates for the counting function are established. The kernel $K(X,Y,Z)$ is supposed to be smooth in $X,Y$ and in $Z\ne 0$ and to admit an asymptotic expansion in homogeneous functions in $Z$ variable as $Z\to 0.$ The order in estimates is determined by the leading homogeneity order in the kernel and geometric properties of the measure $μ$ and involves integral norms of the weight functions $F_1,F_2$. For the case of the measure $μ$ being the surface measure for a Lipschitz surface of some positive codimension $\mathfrak{d},$ in the self-adjoint case, the asymptotics of eigenvalues of this integral operator is found.

math.SP↗
arXiv2022

Commutative algebras of Toeplitz operators on the Bergman space revisited: Spectral theorem approach

For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint operators. We discuss main properties of these representation and, especially, describe relations between properties of the spectral function of Toeplitz operators in the spectral representation and properties of the symbols.

math.FA↗
arXiv2021

Lieb-Thirring estimates for singular measures

Lieb-Thirring type estimates are proved for the sum of powers of negative eigenvalues of a Schrödinger type operator $(-Δ)^l -Vμ$ where $μ$ is a singular measure in $\mathbb{R}^d,$ satisfying a condition on the measure of balls and $V$ is a $μ$-measurable function.

math.SP↗
arXiv2021

Eigenvalues of the Birman-Schwinger operator for singular measures: the noncritical case

In a domain $Ω\subseteq \mathbb{R}^\mathbf{N}$ we consider compact, Birman-Schwinger type, operators of the form $\mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}^*P\mathfrak{A}$; here $P$ is a singular Borel measure in $Ω$ and $\mathfrak{A}$ is a noncritical order $-l\ne -\mathbf{N}/2$ pseudodifferential operator. For a class of such operators, we obtain estimates and a proper version of H.Weyl's asymptotic law for eigenvalues, with order depending on dimensional characteristics of the measure. A version of the CLR estimate for singular measures is proved. For non-selfadjoint operators of the form $P_2 \mathfrak{A} P_1$ and $\mathfrak{A}_2 P \mathfrak{A}_1$ with singular measures $P,P_1,P_2$ and negative order pseudodifferential operators $\mathfrak{A},\mathfrak{A}_1,\mathfrak{A}_2$ we obtain estimates for singular numbers.

math.SP↗
arXiv2021

Eigenvalues of singular measures and Connes noncommutative integration

For a singular measure $μ$, Ahlfors regular of order $α>0,$ with compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator $\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator $\mathbf{T}(P,\mathbf{A}) = \mathbf{A}^*P\mathbf{A}.$ Here $P$ is the signed measure, $P=Vμ$ with density $V$ belonging to the Orlicz class $L^{Ψ,μ}$ with $Ψ(t)=(t+1)\log(t+1)-t.$ Using eigenvalue estimates for such operators, obtained in \texttt{arXiv:2011.14877}, we establish eigenvalue asymptotics of $\mathbf{T}(P,\mathbf{A})$ for a class of measures, including the ones supported on uniformly rectifiable sets. These results lead to the measurability in the sense of A.Connes of operators $\mathbf{T}(P,\mathbf{A})$ and a formula for the singular trace of these operators, producing a noncommutative version of integral with respect to singular measure.

math.SP↗
arXiv2021

Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case

In a domain $Ω\subset \mathbb{R}^{\mathbf{N}}$ we consider a selfadjoint operator $\mathbf{T}=\mathfrak{A}^*P\mathfrak{A} ,$ where $\mathfrak{A}$ is a pseudodifferential operator of order $-l=-\mathbf{N}/2$ and $P=Vμ_Σ$ is a singular signed measure in $Ω$ concentrated on a Lipschitz surface $Σ$ of dimension $d<\mathbf{N}$, absolutely continuous with respect to the surface measure $μ_Σ$ on $Σ$. We establish eigenvalue estimates and asymptotics for this operator. It turns out that the order of these estimates and asymptotics is independent of the dimension $d$ of the surface. If there are several surfaces, possibly, of different dimensions, as well as an absolute continuous measure on $Ω$ the corresponding asymptotic coefficients add up.

math.AP↗