Asymptotics of eigenvalues of the operator describing Aharonov-Bohm effect combined with homogeneous magneticfield coupled with a strong $\delta$-interaction on a loop

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues for a magnetic operator with Aharonov-Bohm and homogeneous magnetic fields combined with a strong delta interaction on a loop, revealing eigenvalue limits and persistent currents.

Contribution

It provides the first detailed asymptotic analysis of eigenvalues for this combined magnetic and delta-interaction operator, including conditions for persistent currents.

Findings

Asymptotic eigenvalue formulas for large delta interaction strength.

Existence of persistent currents for sufficiently strong delta interaction.

Explicit characterization of the operator's spectral properties under combined magnetic effects.

Abstract

We investigate the two-dimensional magnetic operator $H_{c_0,B,\beta} = {(-i\nabla -A)}^{2}-\beta\delta(.-\Gamma),$ where $\Gamma$ is a smooth loop. The vector potential has the form $A=c_0\bigg(\frac{-y}{{x^2+y^2}}; \frac{x}{{x^2+y^2}} \bigg)+ \frac{B}{2}\bigg(-y; x\bigg) $; $B>0,$ $c_0\in]0;1[$. The asymptotics of negative eigenvalues of $H_{c_0,B,\beta}$ for $\beta \longrightarrow +\infty$ is found. We also prove that for large enough positive value of $\beta$ the system exhibits persistent currents.