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For equivalent minima with nearest-neighbour hopping magnitude $\Delta$, paths with $r$ right steps and $s$ left steps have $r-s=n-m$. Summing $(\Delta \tau /\hslash {)}^{r+s}/(r!s!)$ and applying the Fourier representation of a Kronecker delta gives the displayed modified Bessel function kernel in a normalized localized-site basis. Its Fourier exponent gives the tight-binding model energy $E(\theta )=E_{\mathrm{well}}-2\Delta \cos \theta$. Position-endpoint kernels can have an additional common local-wavefunction prefactor.

For the new periodic potential, let $\kappa$ be the single-neighbour hopping rate and $\Delta ϵ=\hslash \kappa$. 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 givesInsert the Fourier representation of a Kronecker delta. The two exponential series sum independently, yielding the dilute-hopping lattice propagatorThe 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 $e^{-\omega \tau /2}$: a positive oscillator ground-state energy must decay under $e^{-H\tau /\hslash }$. The other factors and their derivation remain as displayed. The energy $\Delta ϵ$ is the positive nearest-neighbour quantum tunnelling matrix-element magnitude, exponentially small relative to the local well scale in the semiclassical regime.