# Eigenvalue Asymptotics for the Maass Hamiltonian with Decreasing Electric Potentials

### Shin-ichi Shirai

Ritsumeikan University, Shiga, Japan

## 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 $H={(x,y)∣x∈R,y>0}$ 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.

## Cite this article

Shin-ichi Shirai, Eigenvalue Asymptotics for the Maass Hamiltonian with Decreasing Electric Potentials. Publ. Res. Inst. Math. Sci. 41 (2005), no. 2, pp. 435–457

DOI 10.2977/PRIMS/1145475363