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

We investigate the spectral theory of the invariant Landau Hamiltonian ${\text{La}}^{\nu }$ acting on the space ${\mathcal{F}}_{\mathrm{\Gamma },\chi }^{\nu }$ of $(\mathrm{\Gamma },\chi )$-automotphic functions on ${\text{C}}^n$, for given real number $\nu >0$, lattice $\mathrm{\Gamma }$ of ${\text{C}}^n$ and a map $\chi :\mathrm{\Gamma }\to U(1)$ such that the triplet $(\nu ,\mathrm{\Gamma },\chi )$ 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,$isnontrivialifandonlyif$\lambda=l=0,1,2, …$. In such case, ${\mathcal{E}}_{\mathrm{\Gamma },\chi }^{\nu }(l)$ is a finite dimensional vector space whose the dimension is given explicitly. We show also that the eigenspace ${\mathcal{E}}_{\mathrm{\Gamma },\chi }^{\nu }(0)$ associated to the lowest Landau level of ${\text{La}}^{\nu }$ is isomorphic to the space, ${\mathcal{O}}_{\mathrm{\Gamma },\chi }^{\nu }({\text{C}}^n)$, of holomorphic functions on ${\text{C}}^n$ satisfying $$g(z+\gamma )=\chi (\gamma )e^{\frac{\nu }{2}\mathrm{\mid }\gamma {\mathrm{\mid }}^2+\nu \text{scal}z,\gamma }g(z),\text{eqno}(\ast )$$ that we can realize also as the null space of the differential operator $\sum\limits_{j=1}\limits^n(\frac{-\partial^2}{\partial z_j\partial \bar z_j} + \nu \bar z_j \frac{\partial}{\partial \bar z_j})$actingon$\mathcal C^\infty$functionson$\C^n$satisfying$(*)$.