# 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

H(0) = _y_2( | 1 | ∂ | - | B | )2 - _y_2 | ∂2 | , |

√ -1 | ∂_x_ | y | ∂_y_2 |

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