Square‐integrable solutions and Weyl functions for singular canonical systems

Abstract

Boundary value problems for singular canonical systems of differential equations of the formare studied in the associated Hilbert space L 2 Δ (ı). With the help of a monotonicity principle for matrix functions their square-integrable solutions are specified. This yields a direct treatment of defect numbers of the minimal relation and simultaneously makes it possible to assign certain boundary values to the elements of the maximal relation induced by the system of differential equations in L 2 Δ (ı). The investigation of boundary value problems for these systems and their spectral theory can be carried out by means of abstract boundary triplet techniques. This paper makes explicit the construction and the properties of boundary triplets and Weyl functions for singular canonical systems. Furthermore, the Weyl functions are shown to have a property similar to that of the classical Titchmarsh-Weyl coefficients for singular Sturm-Liouville operators: they single out the square-integrable solutions of the homogeneous systems of canonical differential equations.

Limit‐point/limit‐circle classification for Hain–Lüst type equations

“…The connection between the Titchmarsh–Weyl coefficient and the square‐integrable solutions has been investigated in for special cases. For singular canonical systems of differential equations the method of boundary triplets has been worked out in offering a functional‐analytic/operator theoretic framework to express square‐integrable solutions via the corresponding Weyl functions as an analog of the use of Titchmarsh–Weyl coefficients for the usual Sturm–Liouville expressions.…”

“…The connection between the Titchmarsh-Weyl coefficient and the squareintegrable solutions has been investigated in [15][16][17][18] for special cases. For singular canonical systems of differential equations the method of boundary triplets has been worked out in [5] offering a functional-analytic/operator theoretic framework to express…”

Boundary Value Problems for Integral Equations With Operator Measures

“…Let L be a relation consisting of the pairs {ỹ,f } ∈ H × H satisfying the condition: for each pair {ỹ,f } there exists a pair {y, f } such that the pairs {ỹ,f }, {y, f } are identical in H × H, and equality (3) holds on [a, b 0 ]. By L we denote the closure of L and we call L the maximal relation generated by equation (3). Generally speaking, relation L is not an operator since function y can happen to be identified with zero in H, while f is non-zero.…”

“…We consider equation (3). Let L be a relation consisting of the pairs {ỹ,f } ∈ H × H satisfying the condition: for each pair {ỹ,f } there exists a pair {y, f } such that the pairs {ỹ,f }, {y, f } are identical in H × H, and equality (3) holds on [a, b 0 ].…”

“…Linear relations and operators generated by such differential equations were considered in many works (see [5], [9], further detailed bibliography can be found, for example, in [3]). We also note that linear relations were first employed in work [10] for the description of extensions of differential operators in terms of boundary conditions.…”