On eigenvalues of the Landau Hamiltonian with a periodic electric potential

TL;DR

The paper proves that for every natural number $m$ there exist nonconstant smooth periodic potentials $V$ with zero mean, depending analytically on a small parameter $\\varepsilon$, such that the Landau Hamiltonian $\\widehat{H}_B+V$ possesses the Landau level $\\lambda=(2m+1)B$ as an eigenvalue of infinite multiplicity. The construction reduces to solving a nonlinear finite-dimensional system built from the matrix $\\widehat{C}^{(m)}$, whose simple, positive spectrum is established via a factorization $\\widehat{C}^{(m)}=\\widehat{\\mathcal{E}}^{(m)}\\widehat{\\mathcal{D}}^{(m)}$ and an associated Lyapunov–Schmidt reduction that preserves even, $T$-periodic modes. The approach yields analytic families $\\tau(\\varepsilon)$ and $\\mathfrak{w}_j(\\varepsilon;\cdot)$, producing leader terms $v_j(t)=2\\varepsilon a_j\cos(\\omega t)+O(\\varepsilon^2)$ and a potential $V(x)=2Bm(1-e^{u_1(\\varepsilon;x_2)})$ with zero mean, ensuring eigenvalue embedding for small $\\varepsilon$ and rational magnetic flux. This extends prior results by giving a general construction for all $m$ and clarifying the mechanism by which periodic perturbations embed Landau levels into the spectrum. The findings have implications for spectral engineering in periodic magnetic systems and demonstrate the feasibility of embedding discrete Landau levels into the spectrum via smooth periodic perturbations.

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.