On eigenvalues of the Landau Hamiltonian with a periodic electric potential

TL;DR

This paper demonstrates that for the Landau Hamiltonian with a periodic electric potential, certain Landau levels can be eigenvalues with infinite multiplicity, achieved through smooth, zero-mean potentials depending analytically on a small parameter.

Contribution

It constructs explicit smooth periodic potentials depending on a small parameter that cause specific Landau levels to become eigenvalues with infinite multiplicity.

Findings

Existence of nonconstant periodic potentials with zero mean

Landau levels as eigenvalues with infinite multiplicity

Construction depends analytically on a small parameter

Abstract

We consider the Landau Hamiltonian $\widehat H_B+V$ on $L^2({\mathbb R}^2)$ with a periodic electric potential $V$. For every $m\in {\mathbb N}$ we prove that there exist nonconstant periodic electric potentials $V\in C^{\infty }({\mathbb R}^2;{\mathbb R})$ with zero mean values that analytically depend on a small parameter $\varepsilon \in {\mathbb R}$ such that the Landau level $(2m+1)B$ is an eigenvalue of the Hamiltonian (of infinite multiplicity) where $B>0$ is a strength of a homogeneous magnetic field.