Solvable Models in Quantum Mechanics

Albeverio, Sergio; Gesztesy, Fritz; Høegh-Krohn, Raphaël; Holden, Helge

doi:10.1090/chel/350

Cited by 578 publications

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“…Indeed, the operator −∂ 2 x − γ δ is a self-adjoint operator on L 2 (R) (see [2,Chapter I.3.1] and Section 2 for details). Precisely, we have Proposition 2.…”

Section: Notationsmentioning

confidence: 99%

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Coz

Fukuizumi

Fibich

et al. 2008

Physica D: Nonlinear Phenomena

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We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in H 1 rad (R) and unstable in H 1 (R) under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

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“…Indeed, the operator −∂ 2 x − γ δ is a self-adjoint operator on L 2 (R) (see [2,Chapter I.3.1] and Section 2 for details). Precisely, we have Proposition 2.…”

Section: Notationsmentioning

confidence: 99%

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Coz

Fukuizumi

Fibich

et al. 2008

Physica D: Nonlinear Phenomena

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“…The operator H δ γ corresponds to the formal expression l δ γ = − d 2 dx 2 − γδ (see [7] for details). Equation (1.2) can be viewed as a model of a singular interaction between nonlinear wave and an inhomogeneity.…”

Section: Introductionmentioning

confidence: 99%

Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Pava¹,

Goloshchapova²

2017

Nonlinear Differ. Equ. Appl.

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Abstract. We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schrö-dinger equation with δ-interaction (attractive and repulsive). A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing wave. This is overcome by the perturbation method, the continuation arguments, and the theory of extensions of symmetric operators.Mathematics Subject Classification (2010). Primary 35Q51, 35J61; Secondary 47E05.

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“…As for rigorous mathematical models of such type, they appeared in the 80's [2]. The mathematical background of the approach is the theory of self-adjoint operator extensions (see, e.g., [3,4] and references in [5]) related to model of point-like potentials in quantum physics. There are a large number of works in the field (see, e.g., [6][7][8][9] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Time dependent quantum graph with loop

Eremin¹,

Grishanov²,

Kostrov³

et al. 2017

Nanosystems: Phys. Chem. Math.

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Solvable Models in Quantum Mechanics

Abstract: Library of Congress Cataloging in Publication DataSolvable models in quantum mechanics.(Texts and monographs in physics) Bibliography: p. Includes index.1. Quantum theory-Mathematical models.

Cited by 578 publications

References 0 publications

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Stability of standing waves for NLS-log equation with $$\varvec{\delta }$$-interaction

Time dependent quantum graph with loop

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