The Heat Kernel of a Weighted Maass-Laplacian with Real Weights

Abstract

We derive closed formula for the heat kernel $K_{\mathbb H, k}$ associated to the Maass-Laplacian operator $D_k$ for any real weight $k$ and prove that the heat kernel $K_{\mathbb H, k}$ is strictly monotone decreasing function of the hyperbolic distance. We derive small time and large time asymptotic formulae for the heat kernel $K_{\mathbb H, k}$ and describe its behavior as a function of the real weight $k$.