An Addendum to Krein's Formula

Abstract

We provide additional results in connection with Krein's formula, which describes the resolvent difference of two self-adjoint extensions A 1 and A 2 of a densely defined closed symmetric linear operatorȦ with deficiency indices n n n ∈ ∪ ∞ . In particular, we explicitly derive the linear fractional transformation relating the operator-valued Weyl-Titchmarsh M-functions M 1 z and M 2 z corresponding to A 1 and A 2 .

Weyl-Titchmarsh M -Function Asymptotics for Matrix-valued Schrödinger Operators

“…Our principal motivation in studying this problem stems from our general interest in operator-valued Herglotz functions (cf. [12], [37], [38], [39], [40], [41], [44], [85]) and its possible applications in the areas of inverse spectral theory and completely integrable systems. More precisely, using higher-order expansions of the type (1.1), one can prove trace formulas for Q(x) and certain higher-order differential polynomials in Q(x), following an approach pioneered in [42] (see also [35], [36]).…”

Weyl–Titchmarsh $M$-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

“…For such problems, we note in the subsequent lemma that for fixed β, varying the boundary data α produces Weyl-Titchmarsh matrices M (z, c, x 0 , α, β) related to each other via linear fractional transformations (see also [47], [52] for a general approach to such linear fractional transformations).…”

On Krein's Formula in the Case of Non-densely Defined Symmetric Operators

“…In a recent paper by Gesztesy et al [11], the authors have revisited Krein's formula associated with self-adjoint extensions of a densely defined symmetric operator. They showed that the coefficients in Krein's formula can be expressed in terms of the classical von Neumann parameterization formulas.…”