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

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.