Exponential Quantum Schrödinger Potentials with Subdominant Corrections and the Hilbert–Pólya Problem: Extended Spectral No-Go Theorems

This work investigates half-line quantum Schrödinger operators with a dominant exponential potential and real subdominant corrections in the context of the Hilbert–Pólya problem. It extends an earlier exact spectral no-go result for purely exponential potentials to a substantially broader family of perturbations. The analysis establishes several independent obstructions based on spectral counting, comparison with Morse potentials, relative trace-class methods, oscillatory asymptotics, regularized determinants, and local singularity analysis. It shows that any exact realization of the normalized Riemann xi function would require a unique calibration of the leading exponential potential and would have to evade strong restrictions on the size, sign, oscillation, and spectral effect of the correction term. The study also demonstrates that the full class of subdominant corrections cannot be excluded by the previously available arguments alone. In particular, the relative perturbation may be compact without belonging to any finite Schatten class, and sufficiently oscillatory corrections can generate nontrivial logarithmic contributions to the spectral determinant. A previously unresolved rapidly oscillating example is excluded, while a narrower residual class with exceptional amplitudes and additional compact perturbations remains undecided. The results are unconditional and do not assume the Riemann hypothesis. They neither prove nor disprove the Riemann hypothesis and do not construct a Schrödinger operator whose spectral determinant equals the Riemann xi function. The main outcome is a sharper delineation of the excluded and still-open sectors within this exponential Schrödinger approach to the Hilbert–Pólya problem.