Weyl asymptotics for functional difference operators with power to quadratic exponential potential

We continue the program first initiated in [Geom. Funct. Anal. 26, 288-305 (2016)] and develop a modification of the technique introduced in that paper to study the spectral asymptotics, namely the Riesz means and eigenvalue counting functions, of functional difference operators $\smash{H_0 = \mathcal F^{-1} M_{\cosh(\xi)} \mathcal F}$ with potentials of the form $\smash{W(x) = \lvert{x\rvert}^pe^{\lvert{x\rvert}^\beta}}$ for either $\beta = 0$ and $p > 0$ or $\beta \in (0, 2]$ and $p \geq 0$. We provide a new method for studying general potentials which includes the potentials studied in [Geom. Funct. Anal. 26, 288-305 (2016)] and [J. Math. Phys. 60, 103505 (2019)]. The proof involves dilating the variance of the gaussian defining the coherent state transform in a controlled manner preserving the expected asymptotics.