Weyl–Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control

Abstract

The paper defines the Weyl-Titchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory that suggests an approach to the study of boundary value problems based on the open systems theory founded in works of M. S. Livšic. Examples of boundary value problems for partial differential equations and calculations of their Weyl-Titchmarsh functions are offered as illustration. In particular, we give an independent derivation of the Weyl-Titchmarsh function for the three dimensional Schrödinger operator introduced by W. O. Amrein and D. B. Pearson. Relationships to the Schrödinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted. Mathematics Subject Classification (2000). Primary 47A48, 47B25, 47F05; Secondary 35J10, 47N70.Keywords. Abstract boundary value problem, Green formula, Weyl-Titchmarsh function, operator colligations, linear systems with boundary control, Schrödinger operator. V. RyzhovComp.an.op.th.

Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators

“…Similar Krein type resolvent formulas can also be found in [9,13,25,26,[47][48][49][50]. The fact that the difference of the resolvents belongs to some von Neumann-Schatten class depending on the dimension of the space is well known and goes back to M.S.…”

“…The minus sign in (1.2) is used for technical reasons. It turns out that the operator function λ → Q(λ) is a generalized Q-function in the sense of Definition 2.2 and an explicit variant of Krein's formula for the resolvents of A D and A N is obtained in Theorem 3.4, see also [9,13,25,26,[47][48][49][50] for more general problems. In particular, in the case n = 2 it follows from results due to M.S.…”

Unified approach to critical-contrast homogenisation with explicit links to time-dispersive media

“…Here in the second equality we take advantage of the fact that Γ soft 0 γ soft (z) = I. Furthermore, using the representation (see [13,34])…”

“…stiff does not depend on ε due to the choice of Γ stiff 1 , see(7) and Section 4.1. On the other hand,[34] provides us with the representation M (τ ),stiff ε z (0) = Π * stiff Π stiff , and hence B…”

Elliptic boundary value problems with λ-dependent boundary conditions

“…We point out that for a constant selfadjoint boundary condition τ the solution operatorà coincides with T τ in (1.3) and the above formula reduces to the well-known Krein formula for canonical selfadjoint extensions in L 2 (Ω) of the minimal operator associated to , cf. [1,8,16,[37][38][39]44,[56][57][58][59][60]. The proof of our main result Theorem 4.2 is based on a coupling technique of ordinary and generalized boundary triples which differs from the methods applied in earlier papers.…”