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

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.