A regular analogue of the Smilansky model: spectral properties

Abstract

We analyze spectral properties of the operator $H=\frac{{\partial }^2}{\partial x^2}-\frac{{\partial }^2}{\partial y^2}+{\omega }^2{y}^2-\lambda y^{2}V(xy)$ in $L^2({\mathbb{R}}^2)$, where $\omega \ne 0$ and $V\ge 0$ is a compactly supported and sufficiently regular potential. It is known that the spectrum of $H$ depends on the one-dimensional Schrödinger operator $L=-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+{\omega }^2-\lambda V(x)$ and it changes substantially as $\inf \sigma (L)$ switches sign. We prove that in the critical case, $\inf \sigma (L)=0$, the spectrum of $H$ is purely essential and covers the interval $[0,\infty )$. In the subcritical case, $\inf \sigma (L)>0$, the essential spectrum starts from $\omega$ and there is a non-void discrete spectrum in the interval $[0,\omega )$. We also derive a bound on the corresponding eigenvalue moments.

typos corrected, a reference added; to appear in Rep. Math. Phys

References in corpus (2)

• A regular version of Smilansky model
• Spectral analysis of a class of Schroedinger operators exhibiting a parameter-dependent spectral transition

Cited by in corpus (2)

• A magnetic version of the Smilansky-Solomyak model
• The Kronig-Penney model in a quadratic channel with $\delta$ interactions. II : Scattering approach