A fundamental system of solutions for an operator differential equation with a boundary condition at infinity

Rofe-Beketov, F. S.; Kholkin, Aleksandr M

doi:10.1007/bf01139932

Cited by 5 publications

(7 citation statements)

References 2 publications

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“…by H α . For a discussion of boundary conditions at infinity, see, for instance, [68], [75], and [90]. Our next goal is to construct the square integrable solutions Y (z, ·) ∈ B(H) of τ Y = zY , z ∈ C\R, the B(H)-valued Weyl-Titchmarsh solutions, under the assumptions that a is a regular endpoint for τ and b is of limit-point type for τ .…”

Section: H⊕hmentioning

confidence: 99%

“…Henceforth, under the assumptions of Theorem 3.10, we denote the operator H in L 2 ((a, b); dx; H) associated with the boundary condition induced by α = α * ∈ B(H), that is, the restriction of H max to the set dom(H α ) = {u ∈ dom(H max ) | sin(α)u ′ (a) + cos(α)u(a) = 0} (3.36) by H α . For a discussion of boundary conditions at infinity, see, for instance, [68], [75], and [90].…”

Section: H⊕hmentioning

confidence: 99%

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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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Section: H⊕hmentioning

confidence: 99%

Section: H⊕hmentioning

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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show abstract

“…by H α . For a discussion of boundary conditions at infinity, see, for instance, [85], [92], and [107]. Our next goal is to construct the square integrable solutions Y (z, ·) ∈ B(H) of τ Y = zY , z ∈ C\R, the B(H)-valued Weyl-Titchmarsh solutions, under the assumptions that a is a regular endpoint for τ and b is of limit-point type for τ .…”

Section: Half-line Weyl-titchmarsh Theory For Schrödinger Operators Wmentioning

confidence: 99%

“…by H α . For a discussion of boundary conditions at infinity, see, for instance, [85], [92], and [107].…”

Section: Half-line Weyl-titchmarsh Theory For Schrödinger Operators W...mentioning

confidence: 99%

On spectral theory for Schrödinger operators with operator-valued potentials

Gesztesy

Weikard

Zinchenko

2013

Journal of Differential Equations

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Given a complex, separable Hilbert space H, we consider differential expressions of the type τ = −(d 2 /dx 2 ) + V (x), with x ∈ (a, ∞) or x ∈ R. Here V denotes a bounded operator-valued potential V (·) ∈ B(H) such that V (·) is weakly measurable and the operator norm V (·) B(H) is locally integrable.We consider self-adjoint half-line L 2 -realizations Hα in L 2 ((a, ∞); dx; H) associated with τ , assuming a to be a regular endpoint necessitating a boundary condition of the type sin(α)u ′ (a) + cos(α)u(a) = 0, indexed by the selfadjoint operator α = α * ∈ B(H). In addition, we study self-adjoint full-line L 2 -realizations H of τ in L 2 (R; dx; H). In either case we treat in detail basic spectral theory associated with Hα and H, including Weyl-Titchmarsh theory, Green's function structure, eigenfunction expansions, diagonalization, and a version of the spectral theorem.

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“…In[25,26] a fundamental solution of the equation(3.4) is defined as an operator function (4.6) obeying the conditions (i)-(iii) of Definition 4.3 and, in addition, a selfadjoint boundary condition at the point b (which exists only in the case n b+ = n b− ). In this connection note that statements of Theorems 4.6 and 4.7 complement and generalize similar results obtained in[25,26].4.2.Resolvents of proper extensions of the minimal operator. Let as before Π = {H 0 ⊕ H 1 , Γ 0 , Γ 1 } be a decomposing D-triplet (3.12) for L, let θ = {(C 0 , C 1 ); K} ∈ C(H 0 , H 1 ) be an operator pair (4.1) and let A ∈ Ext L 0 be the corresponding extension(4.3).…”

mentioning

confidence: 99%

Description of generalized resolvents and characteristic matrices of differential operators in terms of the boundary parameter

Mogilevskii¹

2011

Math Notes

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The main objects of our considerations are differential operators generated by a formally selfadjoint differential expression of an even order on the interval [0, b (b ≤ ∞) with operator valued coefficients. We complement and develop the known Shtraus' results on generalized resolvents and characteristic matrices of the minimal operator L 0 . Our approach is based on the concept of a decomposing boundary triplet which enables to establish a connection between the Straus' method and boundary value problems (for singular differential operators) with a spectral parameter in a boundary condition. In particular we provide a parametrization of all characteristic matrices Ω(λ) of the operator L 0 immediately in terms of the Nevanlinna boundary parameter τ (λ). Such a parametrization is given in the form of the block-matrix representation of Ω(λ) as well as by means of the formula for Ω(λ) similar to the well known Krein-Naimark formula for generalized resolvents. In passing we obtain the representation of canonical and generalized resolvents in the form of integral operators with the operator kernel (the Green function) defined in terms of fundamental operator solutions of the equation l[y] − λ y = 0.

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A fundamental system of solutions for an operator differential equation with a boundary condition at infinity

Cited by 5 publications

References 2 publications

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

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

On spectral theory for Schrödinger operators with operator-valued potentials

Description of generalized resolvents and characteristic matrices of differential operators in terms of the boundary parameter

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