Automorphic Dirac Operators and an Analogue of the Quantum Harmonic Oscillator on $Γ\backslash \mathfrak{H}$

Abstract

In the 2010's Bombieri and Garrett showed that the discrete spectrum, if any, of a self-adjoint extension of a restriction of the hyperbolic Laplacian on the upper half plane is $s(s-1)$ where $ζ_{\mathbb{Q}(i)}(s) = 0$. However, they also showed that at most 94% of the zeros of zeta can participate in this spectrum. In an effort to obtain a larger discrete spectrum, Sands in 2020 characterized an automorphic Hamiltonian that is an analogue of the Quantum Harmonic Oscillator for the modular curve. The motivation for this is that the Quantum Harmonic Oscillator is a discretized version of the Laplacian on $\mathbb{R}$. In describing this operator, it is necessary to understand automorphic Dirac operators and the growth of the potential. Here, we explain several noteworthy features of automorphic Dirac operators, prove that Sands' Hamiltonian has a purely discrete spectrum using an explicit presentation of Kronecker's limit formula, and use the operator to give a characterization of an automorphic Schwartz space.