Eigenvalue asymptotics for the Schrödinger operators on the hyperbolic plane

In this paper we consider the Schrödinger operator H_V =-1/2 Δ_ℍ + V on the hyperbolic plane ℍ = {z = (x,y) x∈ ℝ, y > 0}, where Δ_ℍ is the hyperbolic Laplacian and V is a scalar potential on ℍ. It is proven that, under an appropriate condition on V at 'infinity', the number of eigenvalues of H_V less than λ is asymptotically equal to the canonical volume of the quasi-classically allowed region {(x,y; ξ,η) ∈T*ℍ y²(ξ² + η ²)/2 + V(x,y)<λ} as λ→∞. Our proof is based on the probabilistic methods and the standard Tauberian argument as in the proof of Theorem 10.5 in Simon (Functional Integration and Quantum Physics, Academic Press, New York, 1979).