Aharonov and Bohm vs. Welsh eigenvalues

Abstract

We consider a class of two-dimensional Schrödinger operator with a singular interaction of the $\delta$ type and a fixed strength $\beta$ supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov-Bohm flux $\alpha \in [0,\frac{1}{2}]$ in the center. It is shown that if $\beta \ne 0$, there is a critical value ${\alpha }_{\mathrm{crit}}\in (0,\frac{1}{2})$ such that the discrete spectrum has an accumulation point when $\alpha <{\alpha }_{\mathrm{crit}}$, while for $\alpha \ge {\alpha }_{\mathrm{crit}}$ the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed $\alpha \in (0,\frac{1}{2})$ and $|\beta |$ small enough.

18 pages, no figures

References in corpus (2)

• On the dense point and absolutely continuous spectrum for Hamiltonians with concentric $\delta$ shells
• Interlaced dense point and absolutely continuous spectra for Hamiltonians with concentric-shell singular interactions