Schrodinger Operators with Rapidly Oscillating Central Potentials

White, Denis A. W.

doi:10.2307/1999044

Cited by 5 publications

(3 citation statements)

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“…A series of works were devoted to studying oscillating potentials of Wigner-Von Neumann type, like ax −δ sin(bx β ), and their generalizations. See [2,3,32] for some of the results and further references.…”

Section: Introductionmentioning

confidence: 99%

WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

Christ

Kiselev

2001

Commun. Math. Phys.

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Consider a Schrödinger operator on L 2 of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a finite sum of terms, each of which has a derivative of some order in L 1 + L p for some exponent p < 2, then an essential support of the the absolutely continuous spectrum equals R + . Almost every generalized eigenfunction is bounded, and satisfies certain WKB-type asymptotics at infinity. If moreover these derivatives belong to L p with respect to a weight |x| γ with γ > 0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one.

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Section: Introductionmentioning

confidence: 99%

WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

Christ

Kiselev

2001

Commun. Math. Phys.

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“…In stark contrast to Krüger's result for the discrete model, for the continuum model with α, β as in Proposition 1.2 (3) White [20] proves that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.…”

Section: Introductionmentioning

confidence: 58%

“…While there have been several contributions towards this problem in the past, a full solution is still missing. It is worth mentioning that the continuum Schrödinger operator analogue of this problem is completely understood with the final cases settled by White almost 40 years ago [20]. For references on the many contributions leading up to the final result, we refer to White's paper.…”

Section: Introductionmentioning

confidence: 99%

Discrete Schrödinger operators with decaying and oscillating potentials

Frank,

Larson

2024

St. Petersburg Math. J.

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The paper is devoted to a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential V ( n ) = λ n − α cos ⁡ ( π ω n β ) V(n)=\lambda n^{-\alpha }\cos (\pi \omega n^\beta ) with 1 > β > 2 α 1>\beta >2\alpha , it is proved that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.

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Scattering theory for Stark Hamiltonians

Jensen

1994

Proc. Indian Acad. Sci. (Math. Sci.)

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Schrodinger Operators with Rapidly Oscillating Central Potentials

Cited by 5 publications

References 0 publications

WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

Discrete Schrödinger operators with decaying and oscillating potentials

Scattering theory for Stark Hamiltonians

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