# Spectral estimates for a periodic fourth-order operator

Результат исследований: Научные публикации в периодических изданияхстатья

9 Цитирования (Scopus)

### Выдержка

The operator H = d4/dt4 + d/dtpd/dt +q with periodic coefficients p, q on the real line is considered. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. The following statements are proved: 1) the endpoints of gaps are periodic or antiperiodic eigenvalues or branch points of the Lyapunov function, and moreover, their asymptotic behavior at high energy is found; 2) the spectrum of H at high energy has multiplicity two; 3) if p belongs to a certain class, then for any q the spectrum of H has infinitely many gaps, and all branch points of the Lyapunov function, except for a finite number of them, are real and negative; 4) if q = 0 and p → 0, then at the beginning of the spectrum there is a small spectral band of multiplicity 4, and its asymptotic behavior is found; the remaining spectrum has multiplicity 2.

Язык оригинала английский 703-736 34 St. Petersburg Mathematical Journal 22 5 https://doi.org/10.1090/S1061-0022-2011-01164-1 Опубликовано - 1 дек 2011

### Отпечаток

Lyapunov functions
Fourth Order
Operator
Multiplicity
Branch Point
Estimate
Lyapunov Function
High Energy
Asymptotic Behavior
Periodic Coefficients
Absolutely Continuous
Real Line
Eigenvalue
Interval

### Предметные области Scopus

• Анализ
• Алгебра и теория чисел
• Прикладная математика

### Цитировать

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abstract = "The operator H = d4/dt4 + d/dtpd/dt +q with periodic coefficients p, q on the real line is considered. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. The following statements are proved: 1) the endpoints of gaps are periodic or antiperiodic eigenvalues or branch points of the Lyapunov function, and moreover, their asymptotic behavior at high energy is found; 2) the spectrum of H at high energy has multiplicity two; 3) if p belongs to a certain class, then for any q the spectrum of H has infinitely many gaps, and all branch points of the Lyapunov function, except for a finite number of them, are real and negative; 4) if q = 0 and p → 0, then at the beginning of the spectrum there is a small spectral band of multiplicity 4, and its asymptotic behavior is found; the remaining spectrum has multiplicity 2.",
keywords = "Periodic differential operator, Spectral asymptotics, Spectral bands",
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В: St. Petersburg Mathematical Journal, Том 22, № 5, 01.12.2011, стр. 703-736.

Результат исследований: Научные публикации в периодических изданияхстатья

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AB - The operator H = d4/dt4 + d/dtpd/dt +q with periodic coefficients p, q on the real line is considered. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. The following statements are proved: 1) the endpoints of gaps are periodic or antiperiodic eigenvalues or branch points of the Lyapunov function, and moreover, their asymptotic behavior at high energy is found; 2) the spectrum of H at high energy has multiplicity two; 3) if p belongs to a certain class, then for any q the spectrum of H has infinitely many gaps, and all branch points of the Lyapunov function, except for a finite number of them, are real and negative; 4) if q = 0 and p → 0, then at the beginning of the spectrum there is a small spectral band of multiplicity 4, and its asymptotic behavior is found; the remaining spectrum has multiplicity 2.

KW - Periodic differential operator

KW - Spectral asymptotics

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