= Dilute-hopping lattice propagator
{title2=$K_{nm}(\tau)=e^{-E_{\rm well}\tau/\hbar}I_{n-m}(2\Delta\tau/\hbar)$}
For equivalent minima with nearest-neighbour hopping magnitude $\Delta$, paths with $r$ right steps and $s$ left steps have $r-s=n-m$. Summing $(\Delta\tau/\hbar)^{r+s}/(r!s!)$ 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 $E(\theta)=E_{\rm well}-2\Delta\cos\theta$. Position-endpoint kernels can have an additional common local-wavefunction prefactor.
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