Landau (Γ,χ)-automorphic functions on \mathbb{C}^n of magnitude ν

Abstract

We investigate the spectral theory of the invariant Landau Hamiltonian $\La^ν$ acting on the space ${\mathcal{F}}_{\Gamma ,\chi }^{\nu }$ 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}}^ν_{Γ,χ}(λ)=\set{f\in {\mathcal{F}}^ν_{Γ,χ}; \La^νf = ν(2λ+n) f}$; $λ\in\C,$ is non trivial if and only if $\lambda =l=0,1,2,...$. In such case, ${\mathcal{E}}_{\Gamma ,\chi }^{\nu }(l)$ is a finite dimensional vector space whose the dimension is given explicitly. We show also that the eigenspace ${\mathcal{E}}_{\Gamma ,\chi }^{\nu }(0)$ associated to the lowest Landau level of $\La^ν$ is isomorphic to the space, ${\mathcal{O}}^ν_{Γ,χ}(\C^n)$, of holomorphic functions on $\C^n$ satisfying $$ g(z+γ) = χ(γ) e^{\frac ν2 |γ|^2+ν\scal{z,γ}}g(z), \eqno{(*)} $$ that we can realize also as the null space of the differential operator $\sum _{j=1}^n(\frac{-{\partial }^2}{\partial z_j\partial {\bar{z}}_j}+\nu {\bar{z}}_j\frac{\partial }{\partial {\bar{z}}_j})$ acting on ${\mathcal{C}}^{\infty }$ functions on $\C^n$ satisfying $(\ast )$.

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