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 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.
REFERENCES
1.
Evans
, W. D.
, Lewis
, R. T.
, Siedentop
, H.
, and Solovej
, J. P.
, “Counting eigenvalues using coherent states with an application to Dirac and Schrödinger operators in the semi-classical limit
,” Ark. Mat.
34
(2
), 265
–283
(1996
).2.
Grassi
, A.
, Hatsuda
, Y.
, and Mariño
, M.
, “Topological strings from quantum mechanics
,” Ann. Henri Poincaré
17
(11
), 3177
–3235
(2016
).3.
Grassi
, A.
and Mariño
, M.
, “A solvable deformation of quantum mechanics
,” SIGMA Symmetry Integrability Geom. Methods Appl.
15
, 025
(2019
).4.
Griffiths
, R. B.
, “A proof that the free energy of a spin system is extensive
,” J. Math. Phys.
5
, 1215
–1222
(1964
).5.
Kashaev
, R.
and Mariño
, M.
, “Operators from mirror curves and the quantum dilogarithm
,” Commun. Math. Phys.
346
(3
), 967
–994
(2016
).6.
Laptev
, A.
, “Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces
,” J. Funct. Anal.
151
(2
), 531
–545
(1997
).7.
Laptev
, A.
, “On the Lieb-Thirring conjecture for a class of potentials
,” in The Maz’ya Anniversary Collection: Volume 2
(Rostock, 1998), Operator Theory: Advances and Applications Vol. 110 (Birkhäuser
, Basel
, 1999
), pp. 227
–234
.8.
Laptev
, A.
, Schimmer
, L.
, and Takhtajan
, L. A.
, “Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves
,” Geom. Funct. Anal.
26
(1
), 288
–305
(2016
).9.
Lieb
, E. H.
and Simon
, B.
, “The Thomas-Fermi theory of atoms, molecules and solids
,” Adv. Math.
23
(1
), 22
–116
(1977
).10.
Lieb
, E. H.
and Thirring
, W. E.
, “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities
,” in Studies in Mathematical Physics
(Princeton University Press
, Princeton
, 1976
), pp. 269
–303
.11.
Simon
, B.
, “Nonclassical eigenvalue asymptotics
,” J. Funct. Anal.
53
(1
), 84
–98
(1983
).12.
Simon
, B.
, Functional Integration and Quantum Physics
, 2nd ed. (AMS Chelsea Publishing
, Providence, RI
, 2005
).© 2019 Author(s).
2019
Author(s)
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