On absence of embedded eigenvalues for schrÖdinger operators with perturbed periodic potentials

Kuchment, Peter; Vainberg, Boris

doi:10.1080/03605300008821568

Cited by 42 publications

(47 citation statements)

References 30 publications

(56 reference statements)

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“…• It is clear from both this paper and [21,22] that question of irreducibility of the Floquet surface (equivalently, of the Fermi surface, modulo natural periodicity) is intimately related to the problem of existence and behavior of embedded eigenvalues and corresponding eigenfunctions. This does not look like an artifact of the techniques used.…”

Section: Remarks and Acknowledgmentsmentioning

confidence: 99%

“…In the case of localized perturbations of a periodic background, absence of embedded eigenvalues is proven for periodic Schrödinger operators in 1D [26,27]. Albeit the same must surely be true in any dimension, the problem in dimensions higher than 1 is hard and only one limited result is known [21,22]. In the discrete (graph) situation, embedded eigenvalues can arise very easily, due to non-trivial graph topology.…”

Section: Lemma 4 If λ Belongs To the Interior Of A Spectral Band Of mentioning

confidence: 99%

“…In fact, examination of the proofs of this text, as well as of [21,22] shows that we do not need complete irreducibility. What truly required, is that every irreducible component of the Floquet variety intersects the torus T n over an n − 1 dimensional set.…”

Section: Remarks and Acknowledgmentsmentioning

confidence: 99%

“…It was shown in [8] that in 2D irreducibility holds for all but a finitely many of values of the spectral parameter λ. Examples of some separable cases when irreducibility has been proven can be found in [3,8,21,22].…”

Section: Lemma 4 If λ Belongs To the Interior Of A Spectral Band Of mentioning

confidence: 99%

“…Using the described above attachment procedure, one can also easily construct examples when a localized perturbation of a periodic operator does embed an eigenvalue into absolutely continuous spectrum, which is also expected to be impossible in the continuous situation (albeit this is completely proven in dimensions one only [26,27] with just a single result in higher dimension available [21,22]). The aim of this paper is to see what can be said about the eigenfunctions corresponding to such embedded eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

Kuchment

Vainberg

2006

Commun. Math. Phys.

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The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z n . It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only).The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schrödinger operators.

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Section: Remarks and Acknowledgmentsmentioning

confidence: 99%

Section: Lemma 4 If λ Belongs To the Interior Of A Spectral Band Of mentioning

confidence: 99%

Section: Remarks and Acknowledgmentsmentioning

confidence: 99%

Section: Lemma 4 If λ Belongs To the Interior Of A Spectral Band Of mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

Kuchment

Vainberg

2006

Commun. Math. Phys.

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Fermi isospectrality for discrete periodic Schrödinger operators

Liu

2023

Comm Pure Appl Math

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Let normalΓ=q1double-struckZ⊕q2double-struckZ⊕…⊕qddouble-struckZ$\Gamma =q_1\mathbb {Z}\oplus q_2 \mathbb {Z}\oplus \ldots \oplus q_d\mathbb {Z}$, where ql∈Z+$q_l\in \mathbb {Z}_+$, l=1,2,…,d$l=1,2,\ldots ,d$, are pairwise coprime. Let normalΔ+V$\Delta +V$ be the discrete Schrödinger operator, where Δ is the discrete Laplacian on double-struckZd$\mathbb {Z}^d$ and the potential V:Zd→double-struckC$V:\mathbb {Z}^d\rightarrow \mathbb {C}$ is Γ‐periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension d≥3$d\ge 3$: If at some energy level, Fermi varieties of two real‐valued Γ‐periodic potentials V and Y are the same (this feature is referred to as Fermi isospectrality of V and Y), and Y is a separable function, then V is separable; If two complex‐valued Γ‐periodic potentials V and Y are Fermi isospectral and both V=⨁j=1rVj$V=\bigoplus _{j=1}^rV_j$ and Y=⨁j=1rYj$Y=\bigoplus _{j=1}^r Y_j$ are separable functions, then, up to a constant, lower dimensional decompositions Vj$V_j$ and Yj$Y_j$ are Floquet isospectral, j=1,2,…,r$j=1,2,\ldots ,r$; If a real‐valued Γ‐potential V and the zero potential are Fermi isospectral, then V is zero. In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption “Fermi isospectrality” with a stronger assumption “Floquet isospectrality”.

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Integral Representations and Liouville Theorems for Solutions of Periodic Elliptic Equations

Kuchment

Pinchover

2001

Journal of Functional Analysis

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The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral representations are analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation (i.e., for generalized eigenfunctions of Laplace operator). In a previous joint work with Y. Pinchover we described all solutions that can be represented as integrals of positive Bloch solutions over the imaginary Fermi surface, with a hyperfunction as a "measure". Here we characterize the class of solutions such that the corresponding hyperfunction is a distribution on the Fermi surface.

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On absence of embedded eigenvalues for schrÖdinger operators with perturbed periodic potentials

Cited by 42 publications

References 30 publications

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

Fermi isospectrality for discrete periodic Schrödinger operators

Integral Representations and Liouville Theorems for Solutions of Periodic Elliptic Equations

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