### Abstract

In this paper we study the heat trace of the magnetic Schrödinger operator (Equation Presented) on the hyperbolic plane ℍ = {z = (x, y)|x ε ℝ, y > 0}. Here a = (a_{1}, a_{2}) is a magnetic vector potential and V is a scalar potential on ℍ. Under some growth conditions on a and V at infinity, we derive an upper bound of the difference Tr e^{-tHV (0)} - Tr e^{-tHV (a)} as t → +0. As a byproduct, we obtain the asymptotic distribution of eigenvalues less than λ as λ → +∞ when V has exponential growth at infinity (with respect to the Riemannian distance on ℍ). Moreover, we obtain the asymptotics of the logarithm of the eigenvalue counting function as λ → +∞ when V has polynomial growth at infinity. In both cases we assume that a is weaker than V in an appropriate sense.

Original language | English |
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Pages (from-to) | 1-46 |

Number of pages | 46 |

Journal | Mathematical Physics Electronic Journal |

Volume | 12 |

Publication status | Published - 2006 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability

### Cite this

*Mathematical Physics Electronic Journal*,

*12*, 1-46.