We consider the difference operator HW = U + U−1 + W, where U is the self-adjoint Weyl operator U = ebP, b > 0, and the potential W is of the form W(x) = x2N + r(x) with NN and |r(x)| ≤ C(1 + |x|2Nɛ) 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.

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
).
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