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 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.