2D Schrödinger operators with singular potentials concentrated near curves

Abstract

We investigate the Schrödinger operators $H_{\epsilon }=-\Delta +W+V_{\epsilon }$ in ${\mathbb{R}}^2$ with the short-range potentials $V_{\epsilon }$ which are localized around a smooth closed curve $\gamma$. The operators $H_{\epsilon }$ 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 $H_{\epsilon }$. 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 $\epsilon \to 0$, reveal a nontrivial connection between spectral properties of $H_{\epsilon }$ and the geometry of $\gamma$.

21 pages, 3 figures

References in corpus (5)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Potential theory, path integrals and the Laplacian of the indicator
• Approximation of Schrödinger operators with $\delta$-interactions supported on hypersurfaces
• Schrödinger operators with $\delta$-interactions supported on conical surfaces
• Discrete spectrum of interactions concentrated near conical surfaces

Cited by in corpus (1)

• On eigenvibrations of branched structures with heterogeneous mass density