Weyl asymptotics for perturbed functional difference operators

We consider the difference operator H_W = U + U^-1 + W, where U is the self-adjoint Weyl operator U = e^-bP, b > 0, and the potential W is of the form W(x) = x^2N + r(x) with NâN and |r(x)| ≤ C(1 + |x|^2N-É ) for some 0 < É ≤ 2N - 1. This class of potentials W includes polynomials of even degree with leading coefficient 1, which have recently been considered in Grassi and Mariño [SIGMA Symmetry Integrability Geom. Methods Appl. 15, 025 (2019)]. In this paper, we show that such operators have discrete spectrum and obtain Weyl-type asymptotics for the Riesz means and for the number of eigenvalues. This is an extension of the result previously obtained in Laptev et al. [Geom. Funct. Anal. 26, 288-305 (2016)] for W = V + ζV^-1, where V = e^2πbx, ζ > 0.