Let Hx be the one-dimensional Morse Hamiltonian
Hx=−2μℏ2dr2d2+De(1−e−a(r−re))2,
with ωe=a2De/μ and dimensionless anharmonicity
x=4Deℏωe.
Subtract the vibrational ground-state energy and scale energies by ℏωe. The exact bound-state excitation energies are
εv(x)=v[1−x(v+1)],v=1,…,N(x),
where N(x)=⌊(2x)−1−1/2⌋ up to the usual endpoint convention and hence N(x)∼(2x)−1 as x→0+.
Construct a canonical relative/renormalized spectral zeta Zx(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
Zx(s)⟶ζ(s)
for ℜs>1 as x→0+. Prove that, after the explicit renormalization, this convergence extends meromorphically and locally uniformly to a connected domain containing the critical strip 0<ℜs<1.
For every simple nontrivial Riemann zero ρ, prove that for all sufficiently small x>0 there is a unique nearby zero sρ(x) of Zx with
sρ(x)→ρ(x→0+),
and determine the first nonzero asymptotic term in
sρ(x)−ρ
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
Re(sρ(x)−ρ)
take both signs infinitely often as ∣Imρ∣→∞; a quantitative lower bound for both signs or a limiting distribution for suitably normalized displacements is an acceptable stronger result.