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

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 $(*)$.Comment: 20 pages. Minor corrections. Scheduled to appear in issue 8 (2008) of "Journal of Mathematical Physics