Spectra of Self-Adjoint Extensions and Applications to Solvable Schrödinger Operators

Brüning, Jochen; Geyler, V. A.; Pankrashkin, Konstantin

doi:10.1142/s0129055x08003249

Cited by 176 publications

(288 citation statements)

References 106 publications

(144 reference statements)

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“…In particular, this allows one to avoid operators A with empty resolvent set (see Corollary 2.19 and relation (2.37)) for which the formula (2.41) has no sense. In the case of an arbitrary A ∈ Σ J α with non-empty resolvent set, the formula (2.41) also remains true if we interpret T as a J α -self-adjoint relation in H (see [12,Theorem 3.22] for a similar result and [6] for the basic definitions of linear relations theory).…”

Section: The Resolvent Formulamentioning

confidence: 98%

On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

Kuzhel¹,

Patsyuck²

2012

OpMath

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Abstract. Let J and R be anti-commuting fundamental symmetries in a Hilbert space H. The operators J and R can be interpreted as basis (generating) elements of the complex Clifford algebra Cl2(J, R) := span{I, J, R, iJR}. An arbitrary non-trivial fundamental symmetry from Cl2(J, R) is determined by the formula J α = α1J + α2R + α3iJR, where α ∈ S 2 . Let S be a symmetric operator that commutes with Cl2(J, R). The purpose of this paper is to study the sets ΣJ α (∀ α ∈ S 2 ) of self-adjoint extensions of S in Krein spaces generated by fundamental symmetries J α (J α -self-adjoint extensions). We show that the sets ΣJ α and ΣJ β are unitarily equivalent for different α, β ∈ S 2 and describe in detail the structure of operators A ∈ ΣJ α with empty resolvent set.

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Section: The Resolvent Formulamentioning

confidence: 98%

On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

Kuzhel¹,

Patsyuck²

2012

OpMath

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“…1,b. As a result, we obtain a model which provides the satisfactory proximity O(h 1/2 (1 + | ln h|) 3 ) instead of the uncomfortable one O((1 + | ln h|) −2 ) within a simplified but comprehensible version of the conventional asymptotic procedure. It should be mentioned that our model involves only one scalar integral characteristic of each junction zone.…”

Section: Motivationsmentioning

confidence: 99%

“…This material is known and is presented in a condensed form, mainly in order to introduce the notation and explain some technicalities used throughout the paper. We refer to the review papers [29], [3] and [21] for a detailed information. If the skeleton Ξ(0) decouples in the limit, see fig.…”

Section: Outline Of the Papermentioning

confidence: 99%

Scalar problems in junctions of rods and a plate

2018

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In this work we deal with a scalar spectral mixed boundary value problem in a spacial junction of thin rods and a plate. Constructing asymptotics of the eigenvalues, we employ two equipollent asymptotic models posed on the skeleton of the junction, that is, a hybrid domain. We, first, use the technique of self-adjoint extensions and, second, we impose algebraic conditions at the junction points in order to compile a problem in a function space with detached asymptotics. The latter problem is involved into a symmetric generalized Green formula and, therefore, admits the variational formulation. In comparison with a primordial asymptotic procedure, these two models provide much better proximity of the spectra of the problems in the spacial junction and in its skeleton. However, they exhibit the negative spectrum of finite multiplicity and for these "parasitic" eigenvalues we derive asymptotic formulas to demonstrate that they do not belong to the service area of the developed asymptotic models.Keywords: junction of thin rods and plate, scalar spectral problem, asymptotics, dimension reduction, self-adjoint extensions of differential operators, function space with detached asymptotics MSC: 35B40, 35C20, 74K30.

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“…To describe the self-adjoint extensions of the operator H a we use the boundary triple approach, see e.g. [9,20]. The integration by parts gives and a direct computation gives the equality δ(φ, ψ) = a 2 (φ)a 1 (ψ) − a 1 (φ)a 2 (ψ).…”

Section: Decay Of Eigenfunctionsmentioning

confidence: 99%

“…Furthermore, for any (b 1 , b 2 ) ∈ C 2 there is ψ ∈ D(H * a ) such that a 1 (ψ) = b 1 and a 2 (ψ) = b 2 . In the language of [9], the triple (C, a 1 , a 2 ) is a boundary triple for H a , and any self-adjoint extension of H a is a restriction of H * a to the functions ψ satisfying the boundary condition a 1 (ψ) cos ϑ = a 2 (ψ) sin ϑ, where ϑ is a real-valued parameter.…”

Section: Decay Of Eigenfunctionsmentioning

confidence: 99%

Eigenvalues of Robin Laplacians in infinite sectors

Khalile

Pankrashkin

2017

Mathematische Nachrichten

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Abstract. For α ∈ (0, π), let Uα denote the infinite planar sector of opening 2α, Uα = (x 1 , x 2 ) ∈ R 2 : arg(x 1 + ix 2 ) < α , and T γ α be the Laplacian in L 2 (Uα), T γ α u = −∆u, with the Robin boundary condition ∂ν u = γu, where ∂ν stands for the outer normal derivative and γ > 0. The essential spectrum of T γ α does not depend on the angle α and equals [−γ 2 , +∞), and the discrete spectrum is non-empty iff α < π 2 . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for α ≥ π 6 . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by κ/α with a suitable κ > 0, and the nth eigenvalue En(T γ α ) of T γ α behaves asand admits a full asymptotic expansion in powers of α 2 . The eigenfunctions are exponentially localized near the origin. The results are also applied to δ-interactions on star graphs.

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Spectra of Self-Adjoint Extensions and Applications to Solvable Schrödinger Operators

Cited by 176 publications

References 106 publications

On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

Scalar problems in junctions of rods and a plate

Eigenvalues of Robin Laplacians in infinite sectors

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