Perturbations of embedded eigenvalues for self-adjoint ODE systems

Alexia Papalazarou; Sara Maad Sasane

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Perturbations of embedded eigenvalues for self-adjoint ODE systems

Sara Maad Sasane, Alexia Papalazarou

Functional Analysis · Submitted 7 Jun 2021

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

The authors' main claim We show that this set of small perturbations forms a smooth manifold and we specify its co-dimension.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.04 · mentions code or data availability 0.01.

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We consider a perturbation problem for embedded eigenvalues of a self-adjoint differential operator in $L^2(\mathbb R;\mathbb R^n)$. In particular, we study the set of all small perturbations in an appropriate Banach space for which the embedded eigenvalue remains embedded in the continuous spectrum. We show that this set of small perturbations forms a smooth manifold and we specify its co-dimension. Our methods involve the use of exponential dichotomies, their roughness property and Lyapunov-Schmidt reduction.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider a perturbation problem for embedded eigenvalues of a self-adjoint differential operator in $L^2(\mathbb R;\mathbb R^n)$.	background 0.67	0.67	0.04	asserted 0.71
s1	In particular, we study the set of all small perturbations in an appropriate Banach space for which the embedded eigenvalue remains embedded in the continuous spectrum.	contribution 0.97	0.81	0.08	asserted 0.87
s2	We show that this set of small perturbations forms a smooth manifold and we specify its co-dimension.	result 0.98	0.96	0.73	asserted 1.00
s3	Our methods involve the use of exponential dichotomies, their roughness property and Lyapunov-Schmidt reduction.	method 1.00	0.79	0.04	asserted 0.98

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

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The paper

Original abstract

We consider a perturbation problem for embedded eigenvalues of a self-adjoint differential operator in $L^2(\mathbb R;\mathbb R^n)$. In particular, we study the set of all small perturbations in an appropriate Banach space for which the embedded eigenvalue remains embedded in the continuous spectrum. We show that this set of small perturbations forms a smooth manifold and we specify its co-dimension. Our methods involve the use of exponential dichotomies, their roughness property and Lyapunov-Schmidt reduction.

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1 arXiv paper cite this in 12 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Our main result is that the set of perturbations for which the embedded eigenvalue persists forms a smooth manifold with a specified co-dimension.

Embedded eigenvalues for asymptotically periodic ODE systems · 0.85

Papers citing it · 8 sentences without a confident label

Builds on it 1 paper

This is a generalization of the results of the article [ref] , where the problem was studied in the asymptotically constant case.

Embedded eigenvalues for asymptotically periodic ODE systems · Background · 1.00

Uses its method or tool 1 paper

The methods we use combine those in [ref] with methods from Floquet theory.

Embedded eigenvalues for asymptotically periodic ODE systems · Main result · 0.98

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