Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one

Carlone, Raffaele; Finco, Domenico; Tentarelli, Lorenzo

doi:10.1088/1361-6544/ab1273

Cited by 13 publications

(16 citation statements)

References 46 publications

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“…However, by all the limits obtained before, it suffices to prove that u n → u in L p (R 2 ), in order to get (48). Now, from (26),…”

Section: Proposition 23 (Rearrangement Inequality) For Every Pair Of ...mentioning

confidence: 99%

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Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

Adami¹,

Boni²,

Carlone³

et al. 2021

Preprint

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We investigate the ground states for the focusing, subcritical nonlinear Schrödinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for every positive mass and show a logarithmic singularity at the defect. Moreover, up to a multiplication by a constant phase, they are positive, radially symmetric, and decreasing along the radial direction. In order to overcome the obstacles arising from the uncommon structure of the energy space, that complicates the application of standard rearrangement theory, we move to the study of the minimizers of the action functional on the Nehari manifold and then establish a connection with the original problem. An ad hoc result on rearrangements is given to prove qualitative features of the ground states.

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“…However, by all the limits obtained before, it suffices to prove that u n → u in L p (R 2 ), in order to get (48). Now, from (26),…”

Section: Proposition 23 (Rearrangement Inequality) For Every Pair Of ...mentioning

confidence: 99%

“…Such equation has been studied in one (cfr. [9,16,26,36,37]), two (cfr. [5,6,24]) and three dimensions (cfr.…”

Section: Introductionmentioning

confidence: 99%

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

Adami¹,

Boni²,

Carlone³

et al. 2021

Preprint

Self Cite

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“…is strictly convex, which implies that u ω is stable, or strictly concave, which implies that u ω is unstable. However, by the properties of the standing waves, D (ω) = M (ω) and therefore, recalling (15), one concludes the proof.…”

Section: Focusing Casementioning

confidence: 55%

“…The Nonlinear Schrödinger Equation (NLSE) with concentrated nonlinearity in d = 2 is the subject of several recent papers, finalizing a research program developed over the last twenty years (see [8,3,4,14] for the NLSE with concentrated nonlinearity and also [15] and [12] for the fractional case and the Dirac equation, respectively). Such a research line was originally motivated by some mesoscopic physical models.…”

Section: Introductionmentioning

confidence: 99%

Stability of the standing waves of the concentrated NLSE in dimension two

Adami,

Carlone,

Correggi

et al. 2020

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In this paper we will continue the analysis of two dimensional Schrödinger equation with a fixed, pointwise, nonlinearity started in [2,13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features.

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“…Remark 2.5. One can see that Propositions 2.1 and 2.3 do not exploit the whole regularity of φ λ,0 provided by (9). As mentioned in Section 1.2, the regularity provided by (12) would suffice.…”

Section: Introductionmentioning

confidence: 96%

Complete ionization for a non-autonomous point interaction model in d = 2

Borrelli,

Carlone,

Tentarelli

2021

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We consider the two dimensional Schrödinger equation with time dependent delta potential, which represents a model for the dynamics of a quantum particle subject to a point interaction whose strength varies in time. First, we prove global well-posedness of the associated Cauchy problem under general assumptions on the potential and on the initial datum. Then, for a monochromatic periodic potential (which also satisfies a suitable no-resonance condition) we investigate the asymptotic behavior of the survival probability of a bound state of the time-independent problem. Such probability is shown to have a time decay of order O(t −1 ), up to exponentially fast decaying terms.

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Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one

Cited by 13 publications

References 46 publications

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

Stability of the standing waves of the concentrated NLSE in dimension two

Complete ionization for a non-autonomous point interaction model in d = 2

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