On the heat trace of the magnetic Schrödinger operators on the hyperbolic plane
- Autores: Yuzuru Inahama, Shin-ichi Shirai
- Localización: Mathematical Physics Electronic Journal, ISSN-e 1086-6655, Vol. 12, Nº 4, 2006
- Idioma: inglés
- Enlaces
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Resumen
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.
