2D Schrödinger operators with singular potentials concentrated near curves

ABSTRACT

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

Keywords:

• Schrödinger operator
• singular interaction
• δ potential
• ${\delta }^{\mathrm{\prime }}$-interaction
• interaction on curve
• asymptotics of eigenvalues

2000 Mathematics Subject Classifications:

• Primary 35P05
• Secondary 81Q10
• 81Q15

Disclosure statement

No potential conflict of interest was reported by the author(s).