Weyl asymptotics for perturbed functional difference operators

TY - JOUR

T1 - Weyl asymptotics for perturbed functional difference operators

AU - Laptev, Ari

AU - Schimmer, Lukas

AU - Takhtajan, Leon A.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - We consider the difference operator HW = 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) = x2N + 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 = e2πbx, ζ > 0.

AB - We consider the difference operator HW = 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) = x2N + 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 = e2πbx, ζ > 0.

U2 - 10.1063/1.5093401

DO - 10.1063/1.5093401

M3 - Journal article

SN - 0022-2488

VL - 60

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 10

M1 - 103505

ER -