The Energy-Critical Quantum Harmonic Oscillator

Jao, Casey

doi:10.48550/arxiv.1406.2289

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“…This paper is not the first to implement the credo that to treat critical dispersive equations with broken symmetries, one must first solve the limiting problems where the symmetries are restored. Previous examples include [15,16,17,18,19,24,26,27,31,36]. This is the first model to be studied, however, which retains scale invariance, but loses space translation invariance.…”

Section: Introductionmentioning

confidence: 99%

The energy-critical NLS with inverse-square potential

Killip¹,

Miao²,

Vişan³

et al. 2017

DCDS-A

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Section: Introductionmentioning

confidence: 99%

The energy-critical NLS with inverse-square potential

Killip¹,

Miao²,

Vişan³

et al. 2017

DCDS-A

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“…Since the spectrum of H d is discrete, the linear solution does not decay in this case. As was the case for a geometric confinement, this has little effect on global regularity issues [26,23]. As a result, the asymptotic behavior of (1.5) can be quite rich, but very little is known, apart from the existence of many periodic solutions [4] and the existence of KAM tori when d = 1, if one allows a perturbation by a multiplicative potential [14].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behavior of the Nonlinear Schrödinger Equation with Harmonic Trapping

Hani

Thomann

2015

Comm Pure Appl Math

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We consider the cubic nonlinear Schrödinger equation with harmonic trapping on R D (1 Ä D Ä 5). In the case when all directions but one are trapped (aka "cigar-shaped trap"), we prove modified scattering and construct modified wave operators for small initial and final data, respectively. The asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the NLS equation with full trapping on R D 1 . In the physical dimension D D 3, this system turns out to be exactly the (CR) equation derived by Faou, Germain, and the first author as the large box limit of the resonant NLS equation in the homogeneous (zero potential) setting. The special dynamics of the latter equation, combined with the above modified scattering results, allow us to justify and extend some physical approximations in the theory of Bose-Einstein condensates in cigar-shaped traps.

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The Energy-Critical Quantum Harmonic Oscillator

Cited by 2 publications

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The energy-critical NLS with inverse-square potential

The energy-critical NLS with inverse-square potential

Asymptotic Behavior of the Nonlinear Schrödinger Equation with Harmonic Trapping

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