Translationally invariant bidimensional magnetic Laplacians are considered. Using an improved version of the harmonic approximation, we establish the absence of a point spectrum under various assumptions on the behavior of the magnetic field.
REFERENCES
1.
H.
Cycon
, R.
Froese
, W.
Kirsch
, and B.
Simon
, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry
, Texts and Monographs in Physics (Springer-Verlag
, 1987
).2.
P.
Exner
, T.
Kalvoda
, and M.
Tušek
, “A geometric Iwatsuka type effect in quantum layers
,” J. Math. Phys.
59
(4
), 042105
(2018
).3.
P.
Exner
and H.
Kovařík
, “Magnetic strip waveguides
,” J. Phys. A: Math. Gen.
33
(16
), 3297
–3311
(2000
).4.
B.
Helffer
, Semi-Classical Analysis for the Schrödinger Operator and Applications
, Number 1336 in Lecture Notes in Mathematics (Springer-Verlag Berlin Heidelberg
, 1988
).5.
P. D.
Hislop
, N.
Popoff
, N.
Raymond
, and M. P.
Sundqvist
, “Band functions in the presence of magnetic steps
,” Math. Models Methods Appl. Sci.
26
(1
), 161
–184
(2016
).6.
P. D.
Hislop
and E.
Soccorsi
, “Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields
,” J. Math. Anal. Appl.
422
(1
), 594
–624
(2015
).7.
A.
Iwatsuka
, “Examples of absolutely continuous Schrödinger operators in magnetic fields
,” Publ. Res. Inst. Math. Sci.
21
(2
), 385
–401
(1985
).8.
R.
Jost
and A.
Pais
, “On the scattering of a particle by a static potential
,” Phys. Rev.
82
(2
), 840
–851
(1951
).9.
P.
Miranda
and N.
Popoff
, “Spectrum of the Iwatsuka Hamiltonian at thresholds
,” J. Math. Anal. Appl.
460
(2
), 516
–545
(2018
).10.
M.
Mǎntoiu
and R.
Purice
, “Some propagation properties of the Iwatsuka model
,” Commun. Math. Phys.
188
(3
), 691
–708
(1997
).11.
N.
Raymond
, Bound States of the Magnetic Schrödinger Operator
, Volume 27 of Tracts in Mathematics (European Mathematical Society
, 2017
).12.
N.
Raymond
and S.
Vũ Ngoc̣
, “Geometry and spectrum in 2D magnetic wells
,” Ann. Inst. Fourier
65
(1
), 137
–169
(2015
).13.
M.
Reed
and B.
Simon
, Method of Modern Mathematical Physics: Analysis of Operators
(Academic Press
, 1979
), Vol. IV.14.
J.
Reijniers
and F.
Peeters
, “Snake orbits and related magnetic edge states
,” J. Phys.: Condens. Matter
12
(47
), 9771
(2000
).15.
B.
Simon
, “Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: Asymptotic expansions
,” Ann. Inst. H. Poincaré Sect. A
38
(3
), 295
–308
(1983
), available at http://www.numdam.org/item/AIHPA_1983__38_3_295_0.16.
M.
Tušek
, “On an extension of the Iwatsuka model
,” J. Phys. A: Math. Theor.
49
(36
), 365205
(2016
).17.
D.
Yafaev
, “On spectral properties of translationally invariant magnetic Schrödinger operators
,” Ann. Henri Poincaré
9
(1
), 181
–207
(2008
).© 2019 Author(s).
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