# Landau (Γ,χ)-automorphic functions on mathbb{C}^n of magnitude ν - Mathematics > Spectral Theory

Landau (Γ,χ)-automorphic functions on mathbb{C}^n of magnitude ν - Mathematics > Spectral Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: We investigate the spectral theory of the invariant Landau Hamiltonian$\La^ u$ acting on the space ${\mathcal{F}}^ u {\Gamma,\chi}$ of$(\Gamma,\chi)$-automotphic functions on $\C^n$, for given real number $ u>0$,lattice $\Gamma$ of $\C^n$ and a map $\chi:\Gamma\to U(1)$ such that thetriplet $( u,\Gamma,\chi)$ satisfies a Riemann-Dirac quantization typecondition. More precisely, we show that the eigenspace ${\mathcal{E}}^ u {\Gamma,\chi}(\lambda)=\set{f\in{\mathcal{F}}^ u {\Gamma,\chi}; \La^ u f = u(2\lambda+n) f}$;$\lambda\in\C,$ is non trivial if and only if $\lambda=l=0,1,2,

.$. In suchcase, ${\mathcal{E}}^ u {\Gamma,\chi}(l)$ is a finite dimensional vector spacewhose the dimension is given explicitly. We show also that the eigenspace${\mathcal{E}}^ u {\Gamma,\chi}(0)$ associated to the lowest Landau level of$\La^ u$ is isomorphic to the space, ${\mathcal{O}}^ u {\Gamma,\chi}(\C^n)$,of holomorphic functions on $\C^n$ satisfying $$ g(z+\gamma) = \chi(\gamma)e^{\frac u 2 |\gamma|^2+ u\scal{z,\gamma}}g(z), \eqno{(*)} $$ that we canrealize also as the null space of the differential operator$\sum\limits {j=1}\limits^n(\frac{-\partial^2}{\partial z j\partial \bar z j} + u \bar z j \frac{\partial}{\partial \bar z j})$ acting on $\mathcal C^\infty$functions on $\C^n$ satisfying $(*)$.

Author: ** Allal Ghanmi, Ahmed Intissar**

Source: https://arxiv.org/