MPEJ Volume 12, No. 4, 46 pp.
Received: Jun 5, 2005. Accepted:  Aug 7, 2006.

Y. Inahama, S. Shirai
On the heat trace of the magnetic Schrodinger operators on the hyperbolic
plane

ABSTRACT: In this paper we study the heat trace of  the magnetic 
Schr\"{o}dinger operator 
\begin{eqnarray*}
H_{V}(\va) =\frac{1}{2}y^{2}
\left(\frac{1}{\sqrt{-1}}\frac{\partial}{\partial x} - a_{1}(x,y)\right)^{2}
+ \frac{1}{2}y^{2}
\left(\frac{1}{\sqrt{-1}}\frac{\partial}{\partial y} - a_{2}(x,y)\right)^{2}
+V(x,y)
\end{eqnarray*}
on the hyperbolic plane ${\mathbb H}=\{z=(x,y)|x \in {\mathbb R}, y>0\}$.
Here ${\bf a}=(a_{1}, a_{2})$ is a magnetic vector potential and  $V$ is a 
scalar potential on ${\mathbb H}$.
Under some  growth conditions on $\va$ and $V$ at infinity,   
we derive an upper bound of the difference 
${\rm Tr} \,e^{-tH_{V}({\bf 0})}-{\rm Tr}\,e^{-tH_{V}({\va})}$ as  $t \to +0$. 

As a byproduct, we obtain the asymptotic distribution  of eigenvalues less
than $\lambda$ as $\lambda \to + \infty$ when $V$ has  exponential growth at
infinity (with respect to the Riemannian distance on ${\mathbb H}$). 
Moreover, we obtain  the asymptotics of the logarithm of the eigenvalue
counting function  as $\lambda \to + \infty$ when $V$ has polynomial growth at
infinity. 
In both cases we assume that  $\va$ is weaker than $V$ in an appropriate sense.

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