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On absence of embedded eigenvalues for Schr\"{o}dinger operators with perturbed periodic potentials

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Peter Kuchment, Boris Vainberg

Mathematical Physics · Submitted 16 Apr 1999

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

The authors' main claim Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries.

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arXiv IDmath-ph/9904016 (version 1)
Pith IDA5WZNG3Y, Pith's permanent record of this paper and its review
CitationBoris Vainberg, Peter Kuchment. "On absence of embedded eigenvalues for Schr\"{o}dinger operators with perturbed periodic potentials." arXiv:math-ph/9904016 (1999). https://pith.science/paper/A5WZNG3Y
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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, 4,292 input tokens.

What the abstract itself states, as probabilities: claims first 0.55 · states limitation 0.86 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.93 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schr\"{o}dinger operator with a periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schr\"{o}dinger operator with a separable periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential has no embedded eigenvalues.

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 The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schr\"{o}dinger operator with a periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential. background 0.82 0.38 0.09 unclear 0.54
s1 Results of this type have previously been known for the one-dimensional case only. gap 0.75 0.03 0.02 asserted 1.00
s2 Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. result 0.97 0.91 0.85 asserted 1.00
s3 It is conjectured that all periodic potentials satisfy this condition. interpretation 0.56 0.20 0.04 speculative 1.00
s4 Separable periodic potentials satisfy it, and hence in dimensions two and three Schr\"{o}dinger operator with a separable periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential has no embedded eigenvalues. result 0.91 0.72 0.27 asserted 1.00

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

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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

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schr\"{o}dinger operator with a periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schr\"{o}dinger operator with a separable periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential has no embedded eigenvalues.

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