The Abstract Titchmarsh-Weyl M-function for Adjoint Operator Pairs and its Relation to the Spectrum

Brown, Malcolm; Hinchcliffe, James; Marletta, Marco; Naboko, Serguei; Wood, Ian

doi:10.1007/s00020-009-1668-z

Cited by 36 publications

(53 citation statements)

References 21 publications

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“…Other versions of the Krein-type formula as well as its applications to boundary value problems can also be found in [34,35,1,65,13]. The proof of the following result relies on formula (2.5).…”

Section: Kreȋn-type Formula For Resolvents and Comparabilitymentioning

confidence: 93%

Trace formulas for additive and non-additive perturbations

Malamud

Neidhardt

2015

Advances in Mathematics

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Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair {H , H} of maximal accumulative operators has been proved. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H and H = H +V are maximal accumulative and V is trace class, we prove the existence of a summable complex-valued spectral shift function. We also obtain trace formulas for pairs {H, H * } assuming only that H and H * are resolvent comparable. In this case the determinant of the characteristic function of H is involved in trace formulas. The results rely on the technique of boundary triplets for densely and non-densely defined closed symmetric operators. The approach allows to express the spectral shift function of a pair of extensions in terms of the abstract Weyl function and the boundary operator. We improve and generalize certain classical results of M.G. Kreȋn for pairs of self-adjoint and dissipative operators, results of A.V. Rybkin for such pairs as well as of

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Section: Kreȋn-type Formula For Resolvents and Comparabilitymentioning

confidence: 93%

Trace formulas for additive and non-additive perturbations

Malamud

Neidhardt

2015

Advances in Mathematics

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“…Furthermore, the reader is referred to [5]- [7], [9], [10], [12], [20], [21], [30], [53], [55] for some recent extensions and applications of the concept of boundary triplets and their Weyl functions. The aim of this section is to show how boundary triplets for singular canonical systems with equal defect numbers can be chosen and to interpret the corresponding Weyl function as an analytic object that specifies the square-integrable solutions of the underlying homogeneous canonical differential equation.…”

Section: Boundary Triplets and Weyl Functions For Singular Canonical mentioning

confidence: 99%

Square‐integrable solutions and Weyl functions for singular canonical systems

Behrndt

Hassi

Snoo

et al. 2011

Mathematische Nachrichten

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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.

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“…Given the definitions of Γ 1 and Γ 2 above, we have

M (λ) (\begin{matrix} - y^{'} (1) \\ y^{'} (0) \end{matrix}) = (\begin{matrix} y (1) \\ y (0) \end{matrix});

moreover the fact that

(\frac{0pt}{y}) \in ker ({\overset{A}{true}}^{*} - λ I)

yields two equations,

- y^{''} + (q - λ) y + \tilde{w} z = 0, w y + (u - λ) z = 0,

from which z may be eliminated to give the Schur‐complement equation for y , which is

(- \frac{d^{2}}{d x^{2}} + q (x) - λ - \frac{w (x) \overset{w}{true} (x)}{u (x) - λ}) y = 0 .

Thus the Titchmarsh–Weyl function M for the Hain–Lüst operator

{\overset{A}{true}}^{*}

is determined by the formula as applied to any basis of the set of solutions of for λ outside the range of u . Explicit formulae, which we do not require here, are given in [, Eqs. (5.10–5.12)].…”

Section: Preliminariesmentioning

confidence: 99%

“…In recent articles the authors have considered forward and inverse problems for operators in the boundary triples setting. In particular, we have been interested in the detectable subspaces (see below) related to the Titchmarsh–Weyl functions

M (λ)

\tilde{M} (λ)

associated with a formally adjoint pair, which determine upper bounds on the spaces in which the operators can be reconstructed, to some extent, from the information about boundary measurements contained in the Titchmarsh–Weyl functions.…”

Section: Introductionmentioning

confidence: 99%