Minimal spectral functions of an ordinary differential operator

Abstract

Abstract. Let l[y] be a formally selfadjoint differential expression of an even order on the interval [0, b (b ≤ ∞) and let L0 be the corresponding minimal operator. By using the concept of a decomposing boundary triplet we consider the boundary problem formed by the equation l[y] − λy = f (f ∈ L2[0, b ) and the Nevanlinna λ-depending boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the m-function, which in the case of selfadjoint decomposing boundary conditions coincides with the classical characteristic (Titchmarsh-Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e., all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indices n±(L0) and not decomposing boundary conditions) the known estimate of the spectral multiplicity of the (exit space) selfadjoint extension A ⊃ L0. The results of the paper are obtained for expressions l[y] with operator valued coefficients and arbitrary (equal or unequal) deficiency indices n±(L0). IntroductionThe main objects of the paper are differential operators generated by a formally selfadjoint differential expression l[y] of an even order 2n on an interval ∆ = [0, b (b ≤ ∞). We consider the expression l[y] with operator valued coefficients and arbitrary (possibly unequal) deficiency indices, but in order to simplify presentation of the main results assume thatis a scalar expression with real-valued coefficients p k (t) (t ∈ ∆) [20]. Denote by L 0 and L(= L * 0 ) minimal and maximal operators respectively generated by the expression (1.1) in the Hilbert space H := L 2 (∆) and let D be the domain of L. As is known L 0 is a symmetric operator with equal deficiency indices m = n ± (L 0 ) and n ≤ m ≤ 2n. Denote also by n b := m − n the defect number of the expression (1.1) at the point b [17].2000 Mathematics Subject Classification. 34B05, 34B20, 34B40, 47E05. Key words and phrases. 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 formHere 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.Next assume that P = {C 0 (λ), C 1 (λ)} (λ ∈ C \ R) is a Nevanlinna operator pair defined by the block representations3) with the constant entriesĈ 0 ,Ĉ 1 and let τ = τ (λ) := {{h, h ′ } : C 0 (λ)h+C 1 (λ)h ′ = 0} be the corresponding Nevanlinna family of linear relations. Denote byK the range of the operatorĈ = (Ĉ 0Ĉ1 ) and let n = dimK = rank(Ĉ 0Ĉ1 ), n ′ = m −n.Th...

Spectral and pseudospectral functions of various dimensions for symmetric systems

“…For a differential operator l[y] of an even order m formulas similar to (1.6) and (1.7) were proved in our paper [23]. These formulas enable one to calculate spectral functions σ(·) of an arbitrary dimension n σ ( m 2 ≤ n σ ≤ m) corresponding to a special parameter τ ; hence they do not parametrize all spectral functions of l[y].…”

On a generalisation of Krein's example

“…Such concepts allow for example to calculate differences of resolvents of operators with different boundary conditions. There are related works by Boitsev, Neidhardt, and Popov [3] on tensor products of boundary triplets (with bounded operator L), Malamud and Neidhardt [15] for unitary equivalence and regularity properties of different self-adjoint realisations, Gesztesy, Weikard, and Zinchenko [5,6] for a general spectral theory of Schrödinger operators with bounded operator potentials, and Mogilevskii [17], see also the references therein. Moreover, when finishing this paper, the authors of the present paper have learned about the recent paper [2], where Boitsev, Brasche, Malamud, Neidhardt and Popov construct a boundary triplet for the adjoint of the symmetric operator T ⊗ id + id ⊗L with T being symmetric and L being self-adjoint.…”

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

“…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]). Our treatment of spectral theory for halfline and full-line Schrödinger operators in L 2 ((x 0 , ∞); dx; H) and in L 2 (R; dx; H), respectively, in [50], [52] represents the most general one to date.…”