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1D dirac operators with special periodic potentials

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Djakov P., Mityagin B.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

tom 60

nr 1

59-75

For one-dimensional Dirac operators of the form $Ly = i\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} dy/dx + vy$, $v = \begin{pmatrix) 0& Q\\ P & 0\end{pmatrix}$, $y = \begin{pmatrix}y₁\\y₂\end{pmatrix}$, x ∈ ℝ, we single out and study a class X of π-periodic potentials v whose smoothness is determined only by the rate of decay of the related spectral gaps γₙ = |λ⁺ₙ - λ¯ₙ|, where $λₙ^{±}$ are the eigenvalues of L = L(v) considered on [0,π] with periodic (for even n) or antiperiodic (for odd n) boundary conditions.

Eigensystem of an L 2-perturbed harmonic oscillator is an unconditional basis

100%

Adduci J., Mityagin B.

Open Mathematics

2012

tom 10

nr 2

569-589

For any complex valued L p-function b(x), 2 ≤ p < ∞, or L ∞-function with the norm ‖b↾L ∞‖ < 1, the spectrum of a perturbed harmonic oscillator operator L = −d 2/dx 2 + x 2 + b(x) in L 2(ℝ1) is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in L 2(ℝ).

Ograniczanie wyników

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Open Mathematics

2 Mityagin B.

1 Adduci J.

1 Djakov P.

2 2012

Wyniki wyszukiwania

1D dirac operators with special periodic potentials

Eigensystem of an L 2-perturbed harmonic oscillator is an unconditional basis