2D Schr\"{o}dinger operators with singular potentials concentrated near curves

TL;DR

This paper studies how Schrödinger operators with potentials concentrated near a curve in 2D behave asymptotically, revealing connections between spectral properties and the geometry of the curve as the potential becomes singular.

Contribution

It provides a detailed analysis of the eigenvalues and eigenfunctions of approximating Schrödinger operators with singular potentials supported near a curve, linking spectral limits to transmission conditions and geometry.

Findings

Eigenvalues and eigenfunctions exhibit specific asymptotic behavior as the potential concentrates.

Transmission conditions on the curve relate eigenfunctions across the boundary.

Spectral properties are influenced by the geometry of the curve.

Abstract

We investigate the Schr\"{o}dinger operators $H_\varepsilon=-\Delta +W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $\gamma$. The operators $H_\varepsilon$ can be viewed as an approximation of the heuristic Hamiltonian $H=-\Delta+W+a\partial_\nu\delta_\gamma+b\delta_\gamma$, where $\delta_\gamma$ is Dirac's $\delta$-function supported on $\gamma$ and $\partial_\nu\delta_\gamma$ is its normal derivative on $\gamma$. Assuming that the operator $-\Delta +W$ has only discrete spectrum, we analyze the asymptotic behaviour of eigenvalues and eigenfunctions of $H_\varepsilon$. The transmission conditions on $\gamma$ for the eigenfunctions $u^+=\alpha u^-$, $\alpha\, \partial_\nu u^+-\partial_\nu u^-=\beta u^-$, which arise in the limit as $\varepsilon\to 0$, reveal a nontrivial connection between spectral properties of $H_\varepsilon$ and the geometry of $\gamma$.