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

2023/05/10 by Qiu, Yaozhong W.
#34K08 #47A75 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2305.06281

Abstract

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 \smashH₀ = \mathcal F^-1 M_\cosh(ξ) \mathcal F with potentials of the form \smashW(x) = |x|^pe^|x|^β for either β= 0 and p > 0 or β∈ (0, 2] and p ≥ 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.

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