JournalsprimsVol. 41 , No. 2DOI 10.2977/prims/1145475363

Eigenvalue Asymptotics for the Maass Hamiltonian with Decreasing Electric Potentials

  • Shin-ichi Shirai

    Ritsumeikan University, Shiga, Japan
Eigenvalue Asymptotics for the Maass Hamiltonian with Decreasing Electric Potentials cover

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.