A Limiting Absorption Principle for Schrödinger Operators with Generalized Von Neumann–Wigner Potentials II. The Proof

Rejto, Peter; Taboada, Mario

doi:10.1006/jmaa.1997.5294

Cited by 11 publications

(11 citation statements)

References 8 publications

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“…Schrödinger operators with eigenvalues in the continuous spectrum have also been investigated in [3,14,23], and the asymptotic behavior of the solutions of (1.4) for various classes of potentials has also been studied in [4,7,13,15,22]. For perturbations of embedded eigenvalues in situations which are relevant to the automorphic Laplacian and N -body Schrödinger operators see [2,5,9,10,20,25].…”

Section: Introductionmentioning

confidence: 99%

Perturbations of the Wigner-Von Neumann Potential Leaving the Embedded Eigenvalue Fixed

Cruz-Sampedro

Herbst²,

Martínez-Avendaño³

2002

Ann. Henri Poincaré

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We investigate the Schrödinger operator H = −d 2 /dx 2 +(γ/x) sin αx+V , acting in L p (R), 1 ≤ p < ∞, where γ ∈ R \ {0}, α > 0, and V ∈ L 1 (R). For |γ| ≤ 2α/p we show that H does not have positive eigenvalues. For |γ| > 2α/p we show that the set of functions V ∈ L 1 (R), such that H has a positive eigenvalue embedded in the essential spectrum σess(H) = [0, ∞), is a smooth unbounded sub-manifold of L 1 (R) of codimension one. Résumé. On examine l'opérateur de SchrödingerSi |γ| ≤ 2α/p, on montre que H n'a aucune valeur caractéristique positive. Si |γ| > 2α/p, on montre que l'ensemble des fonctions V ∈ L 1 (R), telles que H a une valeur caractéristique positive immergée dans le spectre essentiel σess(H) = [0, ∞), est une sous-variété lisse non-bornée de L 1 (R) de codimensionégaleà un.Proof of Theorem 1.1. For p ≥ 1, α > 0, and γ ∈ R \ {0}, let M p (α, γ) be the set of functions V ∈ L 1 (R) for which the operator H Q,p has a positive eigenvalue. It follows from Theorem 2.1 and Proposition A.1 that

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Section: Introductionmentioning

confidence: 99%

Perturbations of the Wigner-Von Neumann Potential Leaving the Embedded Eigenvalue Fixed

Cruz-Sampedro

Herbst²,

Martínez-Avendaño³

2002

Ann. Henri Poincaré

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“…Then, in Theorem 4.15, we derive a strict, weighted Mourre estimate (1.4) and show that Theorem 1.1 applies, leading to the LAP (1.2) for H 1 . For short range perturbation V , we partially recover results from [DMR,ReT1,ReT2] but, in contrast to these papers, we are able to treat a long range perturbation V . We mention that in [MU], for high enough energies, one proves a LAP for long-range perturbations of a larger class of oscillating potentials.…”

Section: Introductionmentioning

confidence: 64%

“…Our example is a perturbation of a Schrödinger operator with a Wigner-Von Neumann potential. Furthermore we can allow a long range perturbation which is not covered by previous results in [DMR,ReT1,ReT2]. A similar situation is considered in [MU] but at different energies.…”

Section: Introductionmentioning

confidence: 99%

Weighted Mourre's commutator theory, application to Schroedinger operators with oscillating potential

Golenia,

Jecko

2103

J. Operator Theory

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We present a variant of Mourre's commutator theory. We apply it to prove the limiting absorption principle for Schrödinger operators with a perturbed Wigner-Von Neumann potential at suitable energies. To our knowledge, this result is new since we allow a long range perturbation of the Wigner-Von Neumann potential. Furthermore, we can show that the usual Mourre theory, based on differential inequalities and on the generator of dilation, cannot apply to the mentioned Schrödinger operators.

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“…[A], [PSS], [ABG], as well as for decaying oscillatory potentials such as the Wigner-von Neumann potential, see e.g. [DMR], [RT1], [RT2]. Schrödinger operators with Wigner-von Neumann potentials [NW] are of interest because when appropriately calibrated they may produce eigenvalues embedded in the absolutely continuous spectrum of the Hamiltonian, and are linked with the phenomenon of resonance.…”

Section: Introductionmentioning

confidence: 99%

Limiting Absorption Principle for Discrete Schrödinger Operators with a Wigner–von Neumann Potential and a Slowly Decaying Potential

Golenia

Mandich

2020

Ann. Henri Poincaré

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We consider discrete Schrödinger operators on Z d for which the perturbation consists of the sum of a long range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in [Ma1]. To our knowledge, this is a new result even in the one-dimensional case. The improvement consists in a weakening of the assumptions on the long range potential and better LAP weights. The improvement relies only on the fact that the generator of dilations (which serves as conjugate operator) is bounded from above by the position operator. To exploit this, Loewner's theorem on operator monotone functions is invoked.

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A Limiting Absorption Principle for Schrödinger Operators with Generalized Von Neumann–Wigner Potentials II. The Proof

Cited by 11 publications

References 8 publications

Perturbations of the Wigner-Von Neumann Potential Leaving the Embedded Eigenvalue Fixed

Perturbations of the Wigner-Von Neumann Potential Leaving the Embedded Eigenvalue Fixed

Weighted Mourre's commutator theory, application to Schroedinger operators with oscillating potential

Limiting Absorption Principle for Discrete Schrödinger Operators with a Wigner–von Neumann Potential and a Slowly Decaying Potential

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