Weakly coupled bound state of 2D Schrödinger operator with potential-measure

Abstract

We consider a self-adjoint two-dimensional Schrödinger operator $H_{\alpha \mu }$, which corresponds to the formal differential expression \[ -Δ- αμ, \] where $\mu$ is a finite compactly supported positive Radon measure on ${\mathbb{R}}^2$ from the generalized Kato class and $\alpha >0$ is the coupling constant. It was proven earlier that ${\sigma }_{\mathrm{ess}}(H_{\alpha \mu })=[0,+\infty )$. We show that for sufficiently small $\alpha$ the condition $♯{\sigma }_{\mathrm{d}}(H_{\alpha \mu })=1$ holds and that the corresponding unique eigenvalue has the asymptotic expansion $$\lambda (\alpha )=-(C_{\mu }+o(1))\exp (-\frac{4\pi }{\alpha \mu ({\mathbb{R}}^2)}),\alpha \to 0+,$$ with a certain constant $C_{\mu }>0$. We obtain also the formula for the computation of $C_{\mu }$. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend Simon's results, see \cite{Si76}, to the case of potentials-measures. Also for regular potentials our results are partially new.

References in corpus (2)

• Strong coupling asymptotics for Schrödinger operators with an interaction supported by an open arc in three dimensions
• Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc