Abstract

In the first part of this thesis, we derive comparison formulas relating the zeta-regularized determinant of an arbitrary self-adjoint extension of the Laplace operator with domain consisting of smooth functions compactly supported on the complement of a point $P$, to the zeta-regularized determinant of the Laplace operator on $X$. Here $X$ is a compact Riemannian manifold of dimension 2 or 3; $P\in X$. In the second part, we provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an n-dimensional flat torus, the Fourier transform of squares of the eigenfunctions $|phi_j|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda_j$. The thesis is based on two published papers that can be found in the bibliography.