On the boundary value problem for a class of operator-differential equations of odd order with variable coefficients

Алиев, А. Р.

doi:10.1134/s1064562408040078

Cited by 7 publications

(11 citation statements)

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“…A. Dubinskii [9], A. A. Shkalikov [22], S. S. Mirzoev [18], A. R. Aliev [1]- [4], and for the operator-differential equations of the fourth order with multiple characteristics are studied, for example, in the paper by A. R. Aliev [5]. It should be noted that widely enough matters of solvability of operator-differential equations are studied in spaces without weight (see the books by S. G. Krein [16], J.-L. Lions, and E. Magenes [17], V. I. Gorbachuk, and M. L. Gorbachuk [14], J.…”

Section: Introductionmentioning

confidence: 99%

On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space

Алиев

Elbably

2015

Mathematica Slovaca

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In this paper, the conditions of correct solvability are found for a class of the third order operator-differential equations whose principal part has multiple characteristics in the Sobolev type space with exponential weight. The estimations of the norms of intermediate derivative operators closely connected with the solvability conditions are carried out. Moreover, the connection between the weight exponent and the lower bound of the spectrum of the main operator involved in the principal part of the equation is determined in the results of the paper.

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Section: Introductionmentioning

confidence: 99%

On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space

Алиев

Elbably

2015

Mathematica Slovaca

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“…For the particular case of Schrödinger-type operators corresponding to the differential expression τ = −(d 2 /dx 2 )I H + A + V (x) on a bounded interval (a, b) ⊂ R with either A = 0 or A a self-adjoint operator satisfying A ≥ cI H for some c > 0, we refer to the list of references in [52]. For earlier results on various aspects of boundary value problems, spectral theory, and scattering theory in the half-line case (a, b) = (0, ∞), we refer, for instance, to [3], [4], [33], [54]- [56], [57,Chs. 3,4], [58], [60], [64], [78], [80], [93], [96], [98] (the case of the real line is discussed in [100]).…”

Section: Introductionmentioning

confidence: 99%

“…3) and the minimal operator H min in L 2 ((a, b); dx; H) associated with τ ,H min =Ḣ min . H max = (Ḣ min ) * , H * max =Ḣ min = H min .…”

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Donoghue-type m-functions for Schrödinger operators with operator-valued potentials

Gesztesy

Naboko

Weikard

et al. 2019

JAMA

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Given a complex, separable Hilbert space H, we consider differential expressions of the type τ = −(d 2 /dx 2 )I H + V (x), with x ∈ (x 0 , ∞) for some x 0 ∈ R, or x ∈ R (assuming the limit-point property of τ at ±∞). Here V denotes a bounded operator-valued potential V (·) ∈ B(H) such that V (·) is weakly measurable, the operator norm V (·) B(H) is locally integrable, andWe focus on two major cases. First, on m-function theory for self-adjoint half-line L 2 -realizations H +,α in L 2 ((x 0 , ∞); dx; H) (with x 0 a regular endpoint for τ , associated with the selfadjoint boundary condition sin(α)u ′ (x 0 ) + cos(α)u(x 0 ) = 0, indexed by the self-adjoint operator α = α * ∈ B(H)), and second, on m-function theory for self-adjoint full-line L 2 -realizations H of τ in L 2 (R; dx; H).In a nutshell, a Donoghue-type m-function M Do A,N i (·) associated with selfadjoint extensions A of a closed, symmetric operatorȦ in H with deficiency spaces Nz = ker Ȧ * − zI H and corresponding orthogonal projections P Nz onto Nz is given byIn the concrete case of half-line and full-line Schrödinger operators, the role ofȦ is played by a suitably defined minimal Schrödinger operator H +,min in L 2 ((x 0 , ∞); dx; H) and H min in L 2 (R; dx; H), both of which will be proven to be completely non-self-adjoint. The latter property is used to prove that if H +,α in L 2 ((x 0 , ∞); dx; H), respectively, H in L 2 (R; dx; H), are self-adjoint extensions of H +,min , respectively, H min , then the corresponding operator-valued measures in the Herglotz-Nevanlinna representations of the Donoghue-type mfunctions M Do H +,α ,N +,i (·) and M Do H,N i (·) encode the entire spectral information of H +,α , respectively, H.Next, we briefly turn to Donoghue-type m-functions which abstractly can be introduced as follows (cf. [47], [48]). Given a self-adjoint extension A of a densely defined, closed, symmetric operatorȦ in K (a complex, separable Hilbert space) and the deficiency subspace N i ofȦ in K, withthe Donoghue-type m-operator M Do A,Ni (z) ∈ B(N i ) associated with the pair (A, N i ) is given bywith I Ni the identity operator in N i , and P Ni the orthogonal projection in K onto N i . Then M Do A,Ni (·) is a B(N i )-valued Nevanlinna-Herglotz function that admits the representation M DoA,Ni (z) =ˆR dΩ Do A,Ni (λ)

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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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On the boundary value problem for a class of operator-differential equations of odd order with variable coefficients

Cited by 7 publications

References 1 publication

On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space

On a Class of Operator-Differential Equations of the Third Order With Multiple Characteristics on the Whole Axis in the Weighted Space

Donoghue-type m-functions for Schrödinger operators with operator-valued potentials

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

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