Executive Summary : | The hyperbolic Laplacian operator is a linear differential operator on the complex upper half plane H, which is invariant under the matrix group. Harmonic analysis on H focuses on decomposing complex-valued functions into eigenfunctions of the Laplacian operator. The problem of constructing holomorphic or non-holomorphic functions on H has a long history, with Poincare's 1882 Poincare series and Selberg's 1965 non-holomorphic Poincare series playing significant roles in the spectral theory of automorphic functions. These functions are square-integrable on the quotient space and satisfy a differential equation involving the Laplacian operator. The analytic continuation of these Poincare series in the complex variable s to a region greater than half is closely connected to the problem of the existence of eigenvalues of the Laplace operator (so-called exceptional eigenvalues). This project focuses on the weighted Laplacian on the upper half-plane H, which is an analogue of the non-Eucledian Laplacian for non-analytic automorphic forms on the quotient space on H by discrete matrix group and weight k. The spectral theory of the operator has been developed by many mathematicians, and it was proved that the Hilbert space of all square-integrable functions on H transforms in a suitable way with respect to the weight k. The main objective of this project is to construct a family of non-holomorphic Poincare series for the weighted Laplacian, compute their Fourier coefficients, and study interesting analytic properties of such functions in the complex variable s. |