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Grigory Tashchiyan

Publications and source records attributed to Grigory Tashchiyan.

5 recordsLinked to original sources
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↗
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↗
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↗
arXiv2006

On the Spectral Properties of the Perturbed Landau Hamiltonian

The Landau Hamiltonian governing the behavior of a quantum particle in dimension 2 in a constant magnetic field is perturbed by a compactly supported magnetic field and a similar electric field. We describe how the spectral subspaces change and how the Landau levels split under this perturbation.

math-ph↗