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 $H_0={\mathcal{F}}^{-1}{M}_{\cosh (\xi )}\mathcal{F}$ with potentials of the form $W(x)=\mid {x\mid }^p{e}^{\mid {x\mid }^{\beta }}$ for either $\beta =0$ and $p>0$ or $\beta \in (0,2]$ and $p\ge 0$. 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.