Eigenvalue Asymptotics for the Maass Hamiltonian with Decreasing Electric Potentials

Abstract

We study the eigenvalue distribution in the spectral gaps of the Maass Hamiltonian with electric potential $V$. For a real constant $B$, the (unperturbed) Maass Hamiltonian is given by

where $\mathbf{H}=\{(x,y)|x\in \mathbf{R},y>0\}$ is the hyperbolic plane. The spectrum of the Maass Hamiltonian consists of the two disjoint parts: the continuous part and the discrete Landau levels (a finite number of eigenvalues of infinite multiplicity) if $|B|>1/2$. Following the argument as in Raikov, G. D. and Warzel, S. [“Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing potentials”, Rev. Math. Phys., vol. 14, no. 10, (2002), 1051–1072], we obtain the asymptotic distribution of the number of discrete spectrum of $H(V)=H(0)+V$ near each discrete Landau level when $V$ is real-valued, asymptotically spherically symmetric and satisfies some decay estimates near infinity, or $V$ is compactly supported.