Landau (\Gamma,\chi)-automorphic functions on \mathbb{C}^n of magnitude \nu

TL;DR

This paper studies the spectral properties of the Landau Hamiltonian on complex n-dimensional space, characterizing its eigenfunctions, eigenvalues, and the structure of automorphic functions satisfying specific periodicity and holomorphicity conditions.

Contribution

It provides explicit descriptions of eigenspaces, their dimensions, and links the lowest Landau level to holomorphic automorphic functions under a Riemann-Dirac quantization condition.

Findings

Eigenvalues are discrete and correspond to non-negative integers.

Eigenspaces are finite dimensional and explicitly characterized.

Lowest Landau level eigenspace is isomorphic to a space of automorphic holomorphic functions.

Abstract

We investigate the spectral theory of the invariant Landau Hamiltonian $\La^\nu$ acting on the space ${\mathcal{F}}^\nu_{\Gamma,\chi}$ of $(\Gamma,\chi)$-automotphic functions on $\C^n$, for given real number $\nu>0$, lattice $\Gamma$ of $\C^n$ and a map $\chi:\Gamma\to U(1)$ such that the triplet $(\nu,\Gamma,\chi)$ satisfies a Riemann-Dirac quantization type condition. More precisely, we show that the eigenspace $ {\mathcal{E}}^\nu_{\Gamma,\chi}(\lambda)=\set{f\in {\mathcal{F}}^\nu_{\Gamma,\chi}; \La^\nu f = \nu(2\lambda+n) f}$; $\lambda\in\C,$ is non trivial if and only if $\lambda=l=0,1,2, ...$. In such case, ${\mathcal{E}}^\nu_{\Gamma,\chi}(l)$ is a finite dimensional vector space whose the dimension is given explicitly. We show also that the eigenspace ${\mathcal{E}}^\nu_{\Gamma,\chi}(0)$ associated to the lowest Landau level of $\La^\nu$ is isomorphic to the space, ${\mathcal{O}}^\nu_{\Gamma,\chi}(\C^n)$, of holomorphic functions on $\C^n$ satisfying $$ g(z+\gamma) = \chi(\gamma) e^{\frac \nu 2 |\gamma|^2+\nu\scal{z,\gamma}}g(z), \eqno{(*)} $$ that we can realize also as the null space of the differential operator $\sum\limits_{j=1}\limits^n(\frac{-\partial^2}{\partial z_j\partial \bar z_j} + \nu \bar z_j \frac{\partial}{\partial \bar z_j})$ acting on $\mathcal C^\infty$ functions on $\C^n$ satisfying $(*)$.