Designed Pseudo-Laplacians

We elaborate and make rigorous various speculations about the implications of spectral properties of self-adjoint operators on spaces of automorphic forms for location of zeros of $L$-functions. Some of these ideas arose in work of Colin de Verdi\`ere, Lax-Phillips, and Hejhal, from the late 1970s and early 1980s, not to mention semi-apocryphal attributions to P\'olya and Hilbert. For example, given a complex quadratic extension $k$ of $\mathbb Q$, we give a natural self-adjoint extension of a restriction of the invariant Laplacian on the modular curve whose discrete spectrum, if any, consists of values $s(s-1)$ for zeros $s$ of $\zeta_k(s)$. Unfortunately, there seems to be no reason for this discrete spectrum to be large. In fact, Montgomery's pair correlation, and the behavior of $\zeta(1+it)$, imply that at most $94\%$ of zeros of $\zeta(s)$ can appear in this discrete spectrum. Less naively, some preliminary positive results about the dynamics of zeros do follow from these considerations.