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

Abstract

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.

Minimal spectral functions of an ordinary differential operator

“…Proof. According to [19] for each collection τ = {τ + , τ − } ∈ R(H 0 , H 1 ) the corresponding characteristic matrix Ω τ (·) is given by…”

“…Differential operator, decomposing D-boundary triplet, boundary conditions, minimal spectral function, spectral multiplicity. In the present paper we develop an approach based on the concept of a decomposing boundary triplet for a differential operator [17,18,19]. Recall that according to [17] a decomposing boundary triplet for L is a boundary triplet Π = {C n ⊕ C n b , Γ 0 , Γ 1 } in the sense of [9] with the boundary operators Γ j : D → C n ⊕ C n b , j ∈ {0, 1} of the special form…”

“…The problem (1.6)-(1.8) is a particular case of a general Nevanlinna type boundary problem and hence it generates a generalized resolvent R(λ) = R τ (λ) and the corresponding spectral function F (t) = F τ (t) of the operator L 0 [19]. Moreover each selfadjoint boundary problem is given by the boundary condition ( * ) with a constant-valued Nevanlinna pair P = {C 0 , C 1 }, which implies that each canonical resolvent of the operator L 0 is generated by the boundary problem (1.6)-(1.8) with MINIMAL SPECTRAL FUNCTIONS 3 C ′ j1 (λ) ≡ C ′ j1 and C ′ j2 (λ) ≡ C ′ j2 , j ∈ {0, 1}, λ ∈ C \ R. Observe also that the problem (1.6)-(1.8) contains as a particular case a decomposing boundary problem.…”

“…In the present paper we develop an approach based on the concept of a decomposing boundary triplet for a differential operator [17,18,19]. Recall that according to [17] a decomposing boundary triplet for L is a boundary triplet Π = {C n ⊕ C n b , Γ 0 , Γ 1 } in the sense of [9] with the boundary operators Γ j : D → C n ⊕ C n b , j ∈ {0, 1} of the special form Γ 0 y = {y (2) Here y (j) (0) are vectors of quasi-derivatives (2.27) at the point 0 and Γ ′ j y(∈ C n b ), j ∈ {0, 1} are vectors of boundary values of a function y ∈ D at the singular endpoint b.…”

“…Moreover let Ω(λ) = Ω τ (λ) be the Shtraus characteristic matrix of the generalized resolvent R(λ) = R τ (λ) [24]. Then Ω τ (λ) is defined immediately in terms of a Nevanlinna boundary parameter τ by the equalities(see [19]). For a given operator pair (1.4), (1.5) consider also the operator function 11) where…”