We continue the program first initiated by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] 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
=
F
−
1
M
cosh
(
ξ
)
F
H_0 = \mathcal {F}^{-1} M_{\cosh (\xi )} \mathcal {F}
with potentials of the form
W
(
x
)
=
|
x
|
p
e
|
x
|
β
W(x) = \left \lvert {x} \right \rvert ^pe^{\left \lvert {x} \right \rvert ^\beta }
for either
β
=
0
\beta = 0
and
p
>
0
p > 0
or
β
∈
(
0
,
2
]
\beta \in (0, 2]
and
p
≥
0
p \geq 0
. We provide a new method for studying general potentials which includes the potentials studied by Laptev, Schimmer, and Takhtajan [Geom. Funct. Anal. 26 (2016), pp. 288–305] and [J. Math. Phys. 60 (2019), p. 103505]. The proof involves dilating the variance of the gaussian defining the coherent state transform in a controlled manner preserving the expected asymptotics.