Exponential Quantum Schrödinger Potentials and the Hilbert–Pólya Problem: An Exact Spectral No-Go

Abstract

This work investigates a family of one-dimensional quantum Schrödinger operators with purely exponential confining potentials as candidates within the Hilbert–Pólya program. The analysis shows that matching the leading spectral counting behavior associated with the Riemann zeros uniquely fixes the parameters of the exponential potential. The corresponding eigenvalue problem can then be solved in terms of modified Bessel functions, allowing an exact expression for the associated spectral determinant. A direct asymptotic comparison with the Riemann xi function reveals an unavoidable mismatch. This establishes an unconditional no-go result: no operator in the specified purely exponential Schrödinger family with Dirichlet boundary condition reproduces the Riemann xi function as its spectral determinant. The analysis is further extended to constant real Robin boundary conditions, including the Neumann case under strict positivity, where the obstruction remains. The result is limited to this precisely defined family of purely exponential potentials on the half-line. It does not exclude more general Schrödinger potentials, potentials with additional terms, or other operator-theoretic approaches to the Hilbert–Pólya problem.