# P11150: Riemann-Zeta Limit and Zero Motion for the Morse Molecular Bond

- ID: `P11150`
- Reference: `riemann-zeta-limit-zero-motion-morse-molecular-bond`
- Page: https://theoremdb.org/statements/P11150
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/riemann-zeta-limit-zero-motion-morse-molecular-bond?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(H_x\) be the one-dimensional Morse Hamiltonian
\[
H_x=-\frac{\hbar^2}{2\mu}\frac{d^2}{dr^2}+D_e\bigl(1-e^{-a(r-r_e)}\bigr)^2,
\]
with \(\omega_e=a\sqrt{2D_e/\mu}\) and dimensionless anharmonicity
\[
x=\frac{\hbar\omega_e}{4D_e}.
\]
Subtract the vibrational ground-state energy and scale energies by \(\hbar\omega_e\). The exact bound-state excitation energies are
\[
\varepsilon_v(x)=v\,[1-x(v+1)],\qquad v=1,\ldots,N(x),
\]
where \(N(x)=\lfloor (2x)^{-1}-1/2\rfloor\) up to the usual endpoint convention and hence \(N(x)\sim(2x)^{-1}\) as \(x\to0^+\).

Construct a canonical relative/renormalized spectral zeta \(Z_x(s)\) for the full Morse Hamiltonian by combining the bound-state contribution with the scattering spectral-shift contribution relative to the dissociation/free reference Hamiltonian and subtracting explicit semiclassical counterterms. Require the normalization to satisfy
\[
Z_x(s)\longrightarrow \zeta(s)
\]
for \(\Re s>1\) as \(x\to0^+\). Prove that, after the explicit renormalization, this convergence extends meromorphically and locally uniformly to a connected domain containing the critical strip \(0<\Re s<1\).

For every simple nontrivial Riemann zero \(\rho\), prove that for all sufficiently small \(x>0\) there is a unique nearby zero \(s_\rho(x)\) of \(Z_x\) with
\[
s_\rho(x)\to\rho\qquad(x\to0^+),
\]
and determine the first nonzero asymptotic term in
\[
s_\rho(x)-\rho
\]
explicitly from the exact Morse bound-state spectrum and scattering phase. Under RH and simplicity of the nontrivial Riemann zeros, determine whether the horizontal displacements
\[
\operatorname{Re}(s_\rho(x)-\rho)
\]
take both signs infinitely often as \(|\operatorname{Im}\rho|\to\infty\); a quantitative lower bound for both signs or a limiting distribution for suitably normalized displacements is an acceptable stronger result.

## Status

**Resolution status:** Open.
**Lean verification:** Not Lean-verified.

A Lean formalization is attached. Signed verification is pending.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `riemann-zeta-limit-zero-motion-morse-molecular-bond`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.