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?

The paper [1] shows that the eigenfunction of the Schrödinger equation $$ \left(x^2-\frac{1}{4\pi}\frac{d^2}{dx^2}\right)f_n=\frac{2n+1}{2\pi}f_n $$ satisfies the same functional equation as the Riemann zeta function does, and all the zeros of its 1 transform lie on the critical line $\operatorname{Re}(s)=\frac{1}{2}$.
On the other hand, it's also known (see [2]) 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.

Can we get any result in this direction?

References

[1] Daniel Bump, Kwok-Kwong Choi, Pär Kurlberg, Jeffrey Vaaler, "A local Riemann hypothesis. I.", Mathematische Zeitschrift 233, No. 1, 1-19 (2000), DOI:10.1007/PL00004786, MR1738342, Zbl 0991.11022.

[2] Branko Dragovich, "Adelic harmonic oscillator", International Journal of Modern Physics A 10, No. 16, 2349-2365 (1995), MR1334476, Zbl 1044.81585.

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 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 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 mechanics.

Adelic Harmonic Oscillator Branko Dragovich

 https://arxiv.org/abs/hep-th/0404160

Can we get any result in this direction?

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?

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 mechanics.

Adelic Harmonic Oscillator Branko Dragovich

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

Can we get any result in this direction?

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 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 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 mechanics.

Adelic Harmonic Oscillator Branko Dragovich

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

Can we get any result in this direction?