= Solution
For the new periodic potential, let $\kappa$ be the single-neighbour hopping rate and $\Delta\epsilon=\hbar\kappa$. It is determined by the barrier between adjacent minima; an unspecified periodic potential does not determine it from the preceding quartic potential's numerical action. The minimum spacing is now $a$.
A path with $r$ right hops and $s$ left hops has $r-s=n-m$. Summing the ordered-center weights and their direction choices gives
$$
\langle na|e^{-H\tau/\hbar}|ma\rangle\simeq\mathcal A_0e^{-\omega\tau/2}
\sum_{r,s\ge0}\frac{(\kappa\tau)^{r+s}}{r!s!}\delta_{r-s,n-m}.
$$
Insert the <Fourier representation of a Kronecker delta>. The two exponential series sum independently, yielding the <dilute-hopping lattice propagator>
$$
\boxed{\langle na|e^{-H\tau/\hbar}|ma\rangle\simeq\mathcal A_0e^{-\omega\tau/2}
\int_0^{2\pi}\frac{d\theta}{2\pi}e^{-i(n-m)\theta}
\exp\left(\frac{2\Delta\epsilon\tau}{\hbar}\cos\theta\right).}
$$
The two Fourier-sign choices are equivalent by $\theta\mapsto-\theta$. Equivalently the integral is the <modified Bessel function> $I_{n-m}(2\kappa\tau)$.
The original PDF prints $e^{+\omega\tau/2}$ here. Its sign is inconsistent with both the previous double-well result and the requested positive on-site oscillator energy. The correct factor is \b[$e^{-\omega\tau/2}$]: a positive oscillator <ground-state energy> must decay under $e^{-H\tau/\hbar}$. The other factors and their derivation remain as displayed. The energy $\Delta\epsilon$ is the positive nearest-neighbour <quantum tunnelling> matrix-element magnitude, exponentially small relative to the local well scale in the semiclassical regime.
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