Eigenvalue asymptotics for the Schrödinger operators on the hyperbolic plane

Yuzuru Inahama, Shin Ichi Shirai

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


In this paper we consider the Schrödinger operator HV =-1/2 Δ + V on the hyperbolic plane ℍ = {z = (x,y) x∈ ℝ, y > 0}, where Δ 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 HV less than λ is asymptotically equal to the canonical volume of the quasi-classically allowed region {(x,y; ξ,η) ∈T*ℍ y22 + 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).

Original languageEnglish
Pages (from-to)424-456
Number of pages33
JournalJournal of Functional Analysis
Issue number2
Publication statusPublished - Jun 15 2004
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis


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