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 ${\mathcal{F}}^ν_{Γ,χ}$ of $(Γ,χ)$-automotphic functions on $\C^n$, for given real number $ν>0$, lattice $Γ$ of $\C^n$ and a map $χ:Γ\to U(1)$ such that the triplet $(ν,Γ,χ)$ 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 $λ=l=0,1,2, ...$. In such case, ${\mathcal{E}}^ν_{Γ,χ}(l)$ is a finite dimensional vector space whose the dimension is given explicitly. We show also that the eigenspace ${\mathcal{E}}^ν_{Γ,χ}(0)$ 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 $\sum\limits_{j=1}\limits^n(\frac{-\partial^2}{\partial z_j\partial \bar z_j} + ν\bar z_j \frac{\partial}{\partial \bar z_j})$ acting on $\mathcal C^\infty$ functions on $\C^n$ satisfying $(*)$.