The paper below shows the eigenfunction of the Schrödinger equation

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

satisfies the same functional equation as the Riemann 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, by Daniel Bump https://link.springer.com/content/pdf/10.1007/PL00004786.pdf

On the other hand, it's also known that the Rieman zeta function can be considered as a Mellin transform of an eigenfunction of the Schrödinger equation in terms of $p$-adic, adelic quantum mechanics.

Adelic Harmonic Oscillator, by Branko Dragovich https://arxiv.org/abs/hep-th/0404160

Can we get any result in this direction?