TY - JOUR

T1 - Eigenvalue asymptotics for the Schrödinger operators on the hyperbolic plane

AU - Inahama, Yuzuru

AU - Shirai, Shin Ichi

N1 - Funding Information:
·Corresponding author. E-mail addresses: inahama@sigmath.es.osaka-u.ac.jp (Y. Inahama), QZY05360@nifty.ne.jp (S. Shirai). 1Partially supported by JSPS Research Fellowship for Young Scientists.

PY - 2004/6/15

Y1 - 2004/6/15

N2 - In this paper we consider the Schrödinger operator HV =-1/2 Δℍ + V on the hyperbolic plane ℍ = {z = (x,y) x∈ ℝ, y > 0}, where Δℍ 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 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).

AB - In this paper we consider the Schrödinger operator HV =-1/2 Δℍ + V on the hyperbolic plane ℍ = {z = (x,y) x∈ ℝ, y > 0}, where Δℍ 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 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).

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U2 - 10.1016/S0022-1236(03)00237-4

DO - 10.1016/S0022-1236(03)00237-4

M3 - Article

AN - SCOPUS:2942633542

VL - 211

SP - 424

EP - 456

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -