arXiv subjects
Publications and source records attributed to Diana Barseghyan.
The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Dirichlet Laplacian defined on the two-dimensional disk with a radially symmetric magnetic field.
We analyze two-dimensional Schrödinger operators with the potential $|xy|^p - λ(x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $λ\ge 0$, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant $λ$. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for $λ$ below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum covers the positive halfline while the negative spectrum can be only discrete, we demonstrate numerically the existence of a ground state eigenvalue.
We consider a family of quantum graphs $\{(Γ,\mathcal{A}_\varepsilon)\}_{\varepsilon>0}$, where $Γ$ is a $\mathbb{Z}^n$-periodic metric graph and the periodic Hamiltonian $\mathcal{A}_\varepsilon$ is defined by the operation $-\varepsilon^{-1} {\mathrm{d} ^2\over \mathrm{d} x^2}$ on the edges of $Γ$ and either $δ'$-type conditions or the Kirchhoff conditions at its vertices. Here $\varepsilon>0$ is a small parameter. We show that the spectrum of $\mathcal{A}_\varepsilon$ has at least $m$ gaps as $\varepsilon\to 0$ ($m\in\mathbb{N}$ is a predefined number), moreover the location of these gaps can be nicely controlled via a suitable choice of the geometry of $Γ$ and of coupling constants involved in $δ'$-type conditions.
We discuss a modification of Smilansky model in which a singular potential `channel' is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential `channels'.
In this paper we study the eigenvalue sums of Dirichlet Laplacians on bounded domains. Among our results we establish an improvement of the Li-Yau bound in the presence of a constant magnetic field.
We investigate Dirichlet Laplacian in a straight twisted tube of a non-circular cross section, in particular, its discrete spectrum coming from a local slowdown of the twist. We prove a Lieb-Thirring-type estimate for the spectral moments and present two examples illustrating how the bound depends on the tube cross section.
The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.
We analyze two-dimensional Schrödinger operators with the potential $|xy|^p - λ(x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $λ\ge 0$. We show that there is a critical value of $λ$ such that the spectrum for $λ<λ_\mathrm{crit}$ is below bounded and purely discrete, while for $λ>λ_\mathrm{crit}$ it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums.