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research
Spectral properties of a limit-periodic Schr\"odinger operator in dimension two
Authors
H V (x
Young-ran Lee
Yulia Karpeshina
Publication date
26 August 2010
Publisher
View
on
arXiv
Abstract
We study Schr\"{o}dinger operator
H
=
−
Δ
+
V
(
x
)
H=-\Delta+V(x)
H
=
−
Δ
+
V
(
x
)
in dimension two,
V
(
x
)
V(x)
V
(
x
)
being a limit-periodic potential. We prove that the spectrum of
H
H
H
contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves
e
i
⟨
k
⃗
,
x
⃗
⟩
e^{i\langle \vec k,\vec x\rangle }
e
i
⟨
k
,
x
⟩
at the high energy region. Second, the isoenergetic curves in the space of momenta
k
⃗
\vec k
k
corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.Comment: 89 pages, 6 figure
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