  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.

Number Theory