On the heat trace of the magnetic Schrödinger operators on the hyperbolic plane

Yuzuru Inahama, Shin Ichi Shirai

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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 = (a1, a2) 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 languageEnglish
Pages (from-to)1-46
Number of pages46
JournalMathematical Physics Electronic Journal
Volume12
Publication statusPublished - 2006
Externally publishedYes

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All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability

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