Weyl asymptotics for perturbed functional difference operators
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Weyl asymptotics for perturbed functional difference operators. / Laptev, Ari; Schimmer, Lukas; Takhtajan, Leon A.
I: Journal of Mathematical Physics, Bind 60, Nr. 10, 103505, 2019.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Weyl asymptotics for perturbed functional difference operators
AU - Laptev, Ari
AU - Schimmer, Lukas
AU - Takhtajan, Leon A.
PY - 2019
Y1 - 2019
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 N∈ℕ and |r(x)| ≤ C(1 + |x|2N−ɛ) for some 0 < ɛ ≤ 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 N∈ℕ and |r(x)| ≤ C(1 + |x|2N−ɛ) for some 0 < ɛ ≤ 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
VL - 60
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
SN - 0022-2488
IS - 10
M1 - 103505
ER -
ID: 229058070