Around 1915, Pólya and Hilbert independently speculated that self-adjoint operators could explain zeros of L-functions. Reincarnating this idea, Colin de Verdière in the 1980's suggested a physics-oriented way of making inhomogeneous equations into homogeneous ones, i.e. ones with genuine eigenvalues. To make this line of inquiry rigorous, Bombieri and Garrett showed that the simplest instance of Colin de Verdière’s proposed differential equation on Γ \ H does not succeed because the discrete spectrum is small. This motivates understanding analogous operators with a large discrete spectrum. I explain several such automorphic differential operators, discuss how we can understand the Laplacian as a limit of such an operator, and highlight my recent work on these Automorphic Hamiltonians.