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The Initial Value Problem, Scattering and Inverse Scattering, for Non-Linear Schr\"odinger Equations with a Potential and a Non-Local Non-Linearity

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María de los Ángeles Sandoval Romero, Ricardo Weder

Mathematical Physics · Submitted 1 Dec 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Furthermore, we give a method for the unique reconstruction of the potential from the small amplitude limit of the scattering operator.

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arXiv IDmath-ph/0512004 (version 1)
Pith ID5NWZ6PE2, Pith's permanent record of this paper and its review
CitationMar\'ia de los \'Angeles Sandoval Romero, Ricardo Weder. "The Initial Value Problem, Scattering and Inverse Scattering, for Non-Linear Schr\"odinger Equations with a Potential and a Non-Local Non-Linearity." arXiv:math-ph/0512004 (2005). https://pith.science/paper/5NWZ6PE2
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,092 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.31 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail the initial value problem for these equations. In particular, existence and uniqueness of local and global solutions, continuous dependence on the initial data and regularity. We allow for a large class of unbounded potentials. We have no restriction on the growth at infinity of the positive part of the potential. We also construct the scattering operator in the case of potentials that go to zero at infinity. Furthermore, we give a method for the unique reconstruction of the potential from the small amplitude limit of the scattering operator. In the case of the quantum capacitor, our method allows us to uniquely reconstruct all the physical parameters from the small amplitude limit of the scattering operator.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. background 0.98 0.15 0.04 asserted 0.96
s1 We study in detail the initial value problem for these equations. contribution 1.00 0.94 0.04 asserted 0.97
s2 In particular, existence and uniqueness of local and global solutions, continuous dependence on the initial data and regularity. contribution 0.58 0.91 0.10 asserted 0.72
s3 We allow for a large class of unbounded potentials. contribution 0.82 0.80 0.04 asserted 0.89
s4 We have no restriction on the growth at infinity of the positive part of the potential. contribution 0.71 0.82 0.04 asserted 1.00
s5 We also construct the scattering operator in the case of potentials that go to zero at infinity. contribution 0.90 0.94 0.11 asserted 1.00
s6 Furthermore, we give a method for the unique reconstruction of the potential from the small amplitude limit of the scattering operator. contribution 0.80 0.95 0.34 asserted 1.00
s7 In the case of the quantum capacitor, our method allows us to uniquely reconstruct all the physical parameters from the small amplitude limit of the scattering operator. result 0.77 0.93 0.27 asserted 0.93

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

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  • references and citing sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We consider non-linear Schr\"odinger equations with a potential, and non-local non-linearities, that are models in mesoscopic physics, for example of a quantum capacitor, and that also are models of molecular structure. We study in detail the initial value problem for these equations. In particular, existence and uniqueness of local and global solutions, continuous dependence on the initial data and regularity. We allow for a large class of unbounded potentials. We have no restriction on the growth at infinity of the positive part of the potential. We also construct the scattering operator in the case of potentials that go to zero at infinity. Furthermore, we give a method for the unique reconstruction of the potential from the small amplitude limit of the scattering operator. In the case of the quantum capacitor, our method allows us to uniquely reconstruct all the physical parameters from the small amplitude limit of the scattering operator.

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