The paper below shows the eigen function of schrodinger equation

$(x^2-\frac{1}{4\pi}\frac{d^2}{dx^2})f_n=\frac{2n+1}{2\pi}f_n$

satisfies functional equation as Rieman zeta function does and all the zeros of its mellin transform lie on the critical line $Re(s)=\frac{1}{2}$

A local Riemann hypothesis I Daniel Bump
https://link.springer.com/content/pdf/10.1007/PL00004786.pdf

On the other hand, it's also known that rieman zeta function can be considered as a mellin transform of eigen function of schrodinger equation in terms of p-adic, adelic quantum physics.

Adelic Harmonic Oscillator Branko Dragovich

https://arxiv.org/abs/hep-th/0404160


Can we get any result in this direction?