### Abstract

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 y^{2}(ξ^{2} + 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 language | English |
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Pages (from-to) | 424-456 |

Number of pages | 33 |

Journal | Journal of Functional Analysis |

Volume | 211 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 15 2004 |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

*Journal of Functional Analysis*,

*211*(2), 424-456. https://doi.org/10.1016/S0022-1236(03)00237-4

**Eigenvalue asymptotics for the Schrödinger operators on the hyperbolic plane.** / Inahama, Yuzuru; Shirai, Shin Ichi.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 211, no. 2, pp. 424-456. https://doi.org/10.1016/S0022-1236(03)00237-4

}

TY - JOUR

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

AU - Inahama, Yuzuru

AU - Shirai, Shin Ichi

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).

UR - http://www.scopus.com/inward/record.url?scp=2942633542&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2942633542&partnerID=8YFLogxK

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 -