Self-adjoint boundary problems for a second-order differential equation with unbounded operator coefficient

Горбачук, М. Л.

doi:10.1007/bf01075842

Cited by 31 publications

(18 citation statements)

References 1 publication

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“…In accordance with Theorem 4.8(iii) there is a boundary triplet Π Sr = {H Sr , Γ Sr 0 , Γ Sr 1 } for S r * such that H Sr = H Hr ⊗ T = T, Remark 6.1 Sturm-Liouville operators S c with operator-valued potential T = T * ∈ C(T) have first been treated on a finite interval in the pioneering paper by M.L. Gorbachuk [31]. Clearly, the corresponding minimal operator S c admits representation (6.8).…”

Section: Friedrichs and Krein Extensions Of S :=mentioning

confidence: 87%

“…After appearance of the work [31] the spectral theory of self-adjoint and dissipative extensions of S c in L 2 (∆ c , T) has intensively been investigated. The results are summarized in [32,Chapter 4] where one finds, in particular, criteria for discreteness of the spectra, asymptotic formulas for the eigenvalues, resolvent comparability results, etc.…”

Section: Friedrichs and Krein Extensions Of S :=mentioning

confidence: 99%

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Boundary Triplets, Tensor Products and Point Contacts to Reservoirs

Boitsev

Brasche

Malamud

et al. 2018

Ann. Henri Poincaré

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We consider symmetric operators of the form S := A ⊗ I T + I H ⊗ T where A is symmetric and T = T * is (in general) unbounded. Such operators naturally arise in problems of simulating point contacts to reservoirs. We construct a boundary triplet ΠS for S * preserving the tensor structure. The corresponding γ-field and Weyl function are expressed by means of the γ-field and Weyl function corresponding to the boundary triplet ΠA for A * and the spectral measure of T . Applications to 1-D Schrödinger and Dirac operators are given. A model of electron transport through a quantum dot assisted by cavity photons is proposed. In this model the boundary operator is chosen to be the well-known Jaynes-Cumming operator which is regarded as the Hamiltonian of the quantum dot.Mathematics Subject Classification: 47A80, 47B25, 81Q05, 81Q37

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Section: Friedrichs and Krein Extensions Of S :=mentioning

confidence: 87%

Section: Friedrichs and Krein Extensions Of S :=mentioning

confidence: 99%

Boundary Triplets, Tensor Products and Point Contacts to Reservoirs

Boitsev

Brasche

Malamud

et al. 2018

Ann. Henri Poincaré

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“…Rofe Beketov [81], and M.L. Gorbachuk [46] boundary value problems for different classes of differential operators were studied. The following definition was proposed by A. Kochubej [53] (see also [47,67] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Boundary Triplets, Weyl Functions, and the Kreĭn Formula

Derkach

2014

Operator Theory

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This chapter contains a short review of the theory of boundary triplets, and the corresponding Weyl functions, of symmetric operators in Hilbert and Kreȋn spaces. The theory of generalized resolvents of such operators is exposed from the point of view of boundary triplets approach. Applications to different continuation problems related to the extension theory of Kreȋn space symmetric operators are discussed.

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“…For earlier results on various aspects of boundary value problems, spectral theory, and scattering theory in the half-line case (a, b) = (0, ∞), the situation closely related to the principal topic of this paper, we refer, for instance, to [6], [8], [35], [46], [48], [47], [51], [60], [74], [76], [87], [94], [103] (the case of the real line is discussed in [105]). While our treatment of initial value problems was inspired by the one in [94], we permit a more general local behavior of V (·).…”

Section: Introductionmentioning

confidence: 99%

Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials

Gesztesy¹,

Weikard²,

Zinchenko³

2013

Operators and Matrices

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Abstract. We develop Weyl-Titchmarsh theory for self-adjoint Schrödinger operators H α in L 2 ((a,b);dx;H ) associated with the operator-valued differential expression, and H a complex, separable Hilbert space. We assume regularity of the left endpoint a and the limit point case at the right endpoint b . In addition, the bounded self-adjoint operator α = α * ∈ B(H ) is used to parametrize the self-adjoint boundary condition at the left endpoint a of the typewith u lying in the domain of the underlying maximal operator H max in L 2 ((a,b);dx;H ) associated with τ . More precisely, we establish the existence of the Weyl-Titchmarsh solution of H α , the corresponding Weyl-Titchmarsh m -function m α and its Herglotz property, and determine the structure of the Green's function of H α .Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient V ,, and f ∈ L 1 loc ((a,b);dx;H ) . We also study the analog of this initial value problem with y and f replaced by operator-valued functions Y,F ∈ B(H ) .Our hypotheses on the local behavior of V appear to be the most general ones to date. Mathematics subject classification (2010): Primary: 34A12, 34B20, 34B24; Secondary: 47E05.

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Self-adjoint boundary problems for a second-order differential equation with unbounded operator coefficient

Cited by 31 publications

References 1 publication

Boundary Triplets, Tensor Products and Point Contacts to Reservoirs

Boundary Triplets, Tensor Products and Point Contacts to Reservoirs

Boundary Triplets, Weyl Functions, and the Kreĭn Formula

Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials

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