On the eigenvalues of Aharonov–Bohm operators with varying poles

Bonnaillie‐Noël, Virginie; Noris, Benedetta; Nys, Manon; Terracini, Susanna

doi:10.2140/apde.2014.7.1365

Cited by 17 publications

(82 citation statements)

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“…The fact that X is a critical point is proved in the case N = 1 for the minimal 3-partitions of a simply connected domain in [6]. As far as we know, our result is new for a general k-partition.…”

Section: The Map U (T) Is Unitary and Satises U (T)(cmentioning

confidence: 72%

“…During the writing of this work, S. Terracini showed us a preliminary version of the paper [6]. It contains a similar continuity result, restricted to the case of one pole and assuming Ω to be simply connected with a C ∞ -boundary.…”

Section: Nmentioning

confidence: 95%

“…Most of the quoted authors have restricted themselves to the case where no α i is an integer, and the form domain is contained in H 1 (Ω) . This is for example the case in [2,15,6]. This does not introduce any restriction for xed poles, since an integer ux can be brought to zero by a gauge transformation dened on Ω X .…”

Section: Characterization Of the Form Domainmentioning

confidence: 99%

“…Let us assume that λ k (0) is simple and has a K X -real eigenfunction u with at least three nodal lines that meet at X i . Then λ k (0) = 0 .During the writing of this work, S. Terracini showed us a preliminary version of the paper [6]. It contains a similar continuity result, restricted to the case of one pole and assuming Ω to be simply connected with a C ∞ -boundary.…”

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confidence: 92%

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Eigenvalues variations for Aharonov-Bohm operators

Léna¹

2015

Journal of Mathematical Physics

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We study how the eigenvalues of a magnetic Schrödinger operator of Aharonov-Bohm type depend on the singularities of its magnetic potential. We consider a magnetic potential dened everywhere in R 2 except at a nite number of singularities, so that the associated magnetic eld is zero. On a xed planar domain, we dene the corresponding magnetic Hamiltonian with Dirichlet boundary conditions, and study its eigenvalues as functions of the singularities. We prove that these functions are continuous, and in some cases even analytic. We sketch the connection of this eigenvalue problem to the problem of nding spectral minimal partitions of the domain. IntroductionAharonov-Bohm operators have been introduced in [1] as models of Schrödinger operators with a localized magnetic eld. In addition to their physical relevance, it has been shown in [8] that these operators appear in the theory of spectral minimal partitions (see [9] for a denition of the latter object). In [4,3,5], the eigenvalues and eigenfunctions of an Aharonov-Bohm operator with Dirichlet boundary condition have thus been studied numerically to nd minimal partitions. One of the aim of the present work is to support and generalize some observations made in these papers.We dene Aharonov-Bohm operators as follows. Let ω be an open and connected set in R 2 . As usual, we denote by C ∞ c (ω) the set of smooth functions compactly supported in ω. Generally speaking, let us consider a magnetic potential, that is to say a vector eld A ∈ C ∞ (ω, R 2 ). We dene the sesquilinear form s A byand the norm · A by ) . In the rest of the paper, N is an integer, X = (X 1 , . . . , X N ) an N -tuple of points in R 2 , with X i = X j for i = j (at least for now), andWe dene the Aharonov-Bohm potential associated with X and α as the vector eldLet us point out that the X i 's can be in R 2 \ Ω, and in particular in ∂Ω. TheAharonov-Bohm operator associated with X and α is the magnetic Hamiltonianon Ω X . We denote it by −∆ X α , and the associated form domain and quadratic form by Q X α and q X α respectively. Along the paper, we make frequent references to the Dirichlet realization of the Laplacian on Ω , and to the sequence of its eigenvalues. We denote them by −∆Let us conclude by a few remarks of a more physical nature. To an AharonovBohm potential is associated a magnetic eldwhich is a measure. If B i is a small disk centered at X i , such that X j / ∈ B i for j = i ,The coecient α i can therefore be called the normalized (magnetic) ux at X i .Let nally note that for any closed loop γ inwhere ind γ (X i ) is the winding number of γ around X i . We use in this paper the following characterization of the form domain Q X α , which follows from Hardy-type inequalities proved in [12,2,14]. Proposition 1.1. Let H X α be dened aswith the natural scalar product. Then H X α is a Hilbert space compactly embedded in L 2 (Ω). Furthermore, there exists a continuous mapping(Ω) belongs to Q X α if, and only if, u ∈ H X α and γ 0 u = 0 .As a consequence of the compact embedding, the sp...

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Section: The Map U (T) Is Unitary and Satises U (T)(cmentioning

confidence: 72%

Section: Nmentioning

confidence: 95%

Section: Characterization Of the Form Domainmentioning

confidence: 99%

mentioning

confidence: 92%

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Eigenvalues variations for Aharonov-Bohm operators

Léna¹

2015

Journal of Mathematical Physics

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“…Let us fix ε 1 and ε 2 in (0, ε 0 ] such that ε 1 > ε 2 . Since H 1 0 (Ω) ⊂ Q ε2 ⊂ Q ε1 , the definitions given by Formulas (6) and (7) immediately imply that λ j (ε 1 ) ≤ λ j (ε 2 ) ≤ λ j for each integer j ≥ 1. The function (0, ε 0 ] ∋ ε → λ j (ε) is therefore non-increasing and bounded by λ j for each integer j ≥ 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Eigenvalue variation under moving mixed Dirichlet–Neumann boundary conditions and applications

2020

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We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet-Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion's leading term. This allows inferring some remarkable consequences for Aharonov-Bohm eigenvalues when the singular part of the operator has two coalescing poles.

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