The Lyapunov function for Schrödinger operators with a periodic 2×2 matrix potential

Результат исследований: Научные публикации в периодических изданияхстатьянаучнаярецензирование

8 Цитирования (Scopus)

Выдержка

We consider the Schrödinger operator on the real line with a 2×2 matrix-valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two-sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and the anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: (1) stable gaps, where the endpoints are the periodic and the anti-periodic eigenvalues, (2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We also show that periodic and anti-periodic spectrum together determine the spectrum of the matrix Hill operator.
Язык оригиналарусский
Страницы (с-по)106-126
Число страниц21
ЖурналJournal of Functional Analysis
Том234
Номер выпуска1
DOI
СостояниеОпубликовано - 2006

Ключевые слова

  • Schrödinger operator
  • Periodic matrix potentials
  • Spectral bands
  • Spectral gaps

Цитировать

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title = "The Lyapunov function for Schr{\"o}dinger operators with a periodic 2×2 matrix potential",
abstract = "We consider the Schr{\"o}dinger operator on the real line with a 2×2 matrix-valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two-sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and the anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: (1) stable gaps, where the endpoints are the periodic and the anti-periodic eigenvalues, (2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We also show that periodic and anti-periodic spectrum together determine the spectrum of the matrix Hill operator.",
keywords = "Schr{\"o}dinger operator, Periodic matrix potentials, Spectral bands, Spectral gaps",
author = "Andrei Badanin and Jochen Br{\"u}ning and Коротяев, {Евгений Леонидович}",
year = "2006",
doi = "10.1016/j.jfa.2005.11.012",
language = "русский",
volume = "234",
pages = "106--126",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Elsevier",
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The Lyapunov function for Schrödinger operators with a periodic 2×2 matrix potential. / Badanin, Andrei; Brüning, Jochen; Коротяев, Евгений Леонидович.

В: Journal of Functional Analysis, Том 234, № 1, 2006, стр. 106-126.

Результат исследований: Научные публикации в периодических изданияхстатьянаучнаярецензирование

TY - JOUR

T1 - The Lyapunov function for Schrödinger operators with a periodic 2×2 matrix potential

AU - Badanin, Andrei

AU - Brüning, Jochen

AU - Коротяев, Евгений Леонидович

PY - 2006

Y1 - 2006

N2 - We consider the Schrödinger operator on the real line with a 2×2 matrix-valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two-sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and the anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: (1) stable gaps, where the endpoints are the periodic and the anti-periodic eigenvalues, (2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We also show that periodic and anti-periodic spectrum together determine the spectrum of the matrix Hill operator.

AB - We consider the Schrödinger operator on the real line with a 2×2 matrix-valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two-sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and the anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: (1) stable gaps, where the endpoints are the periodic and the anti-periodic eigenvalues, (2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We also show that periodic and anti-periodic spectrum together determine the spectrum of the matrix Hill operator.

KW - Schrödinger operator

KW - Periodic matrix potentials

KW - Spectral bands

KW - Spectral gaps

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DO - 10.1016/j.jfa.2005.11.012

M3 - статья

VL - 234

SP - 106

EP - 126

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

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