Boundary data maps and Krein's resolvent formula for Sturm-Liouville operators on a finite interval

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals \((a,b) \subseteq \mathbb{R}\) associated with rather general differential expressions of the type \begin{equation*}\tau f = \frac{1}{\tau} (-(p[f'+sf])'+sp[f'+sf]+qf),\end{equation*} where the coefficients \(p, q, r, s\) are real-valued and Lebesgue measurable on \((a,b)\), with \(p \neq 0\), \(r > 0\) a.e. on \((a,b)\), and \(p^{-1}, q, r, s \in L_{loc}^1((a,b),dx)\), and \(f\) is supposed to satisfy \begin{equation*} f \in AC_{loc}((a,b)), p[f'+sf] \in AC_{loc}((a,b)). \end{equation*} In particular, this setup implies that \(\tau\) permits a distributional potential coefficient, including potentials in \(H_{loc}^{-1}((a,b))\). We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator \(T_{max}\), or equivalently, all self-adjoint extensions of the minimal operator \(T_{min}\), all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of \(T_{min}\). In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of \(T_{min}\). Finally, in the special case where \(\tau\) is regular, we characterize the Krein-von Neumann extension of \(T_{min}\) and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups)

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“…An elucidation along these lines for the case s = 0 a.e. on (a, b) was set forth in [26]. Theorem 12.3.…”

Section: The Krein-von Neumann Extension In the Regular Casementioning

confidence: 99%

“…Following [26] and [57], the sesquilinear forms associated to (13.1) and (13.2) are readily written down and read (cf. Appendix A)…”

Section: Positivity Preserving and Improving Resolvents And Semigroupmentioning

confidence: 99%

Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

Eckhardt¹,

Gesztesy²,

Nichols³

et al. 2013

OpMath

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139

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“…As shown in[8, Example 3.3], the resulting operator H 0,RK represents the Kreinvon Neumann extension of H min . Thus,…”

mentioning

confidence: 99%

Effective computation of traces, determinants, and ζ-functions for Sturm–Liouville operators

Gesztesy

Kirsten

2019

Journal of Functional Analysis

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Dedicated with great pleasure to Konstantin Makarov on the occasion of his 60th birthday.Abstract. The principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents (and resolvent differences), Fredholm determinants, ζ-functions, and ζ-function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and ζ-function regularized determinants.Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm-Liouville operators on compact intervals with various (separated and coupled) boundary conditions, and Schrödinger operators on a halfline, are provided and further illustrated with an array of examples.2) for 0 < ε < ε 0 and D(z 0 ; ε 0 )\{z 0 } ⊂ ρ(T ); here D(z 0 ; r 0 ) ⊂ C is the open disk with center z 0 and radius r 0 > 0, and C(z 0 ; r 0 ) = ∂D(z 0 ; r 0 ) the corresponding circle. The geometric multiplicity m g (z 0 ; T ) of an eigenvalue z 0 ∈ σ p (T ) is defined by m g (z 0 ; T ) = dim(ker((T − z 0

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“…For the remainder of this section we make the following assumptions: 9) for some N 0 = N 0 (z 1 ) ∈ N and some 0 < ε 0 = ε 0 (z 1 ) sufficiently small, with…”

Section: )mentioning

confidence: 99%

On the index of meromorphic operator-valued functions and some applications

Behrndt¹,

Gesztesy

Holden

et al. 2017

EMS Series of Congress Reports

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Dedicated with great pleasure to Pavel Exner at the occasion of his 70th birthday.Abstract. We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman-Schwinger-type operators and to the operator-valued Weyl-Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided. Contents 1. Introduction 1 2. On Factorizations of Analytic Operator-Valued Functions 3 3. Algebraic Multiplicities of Zeros of Analytic Fredholm Operators 7 4. On the Notion of an Index of Meromorphic Operator-Valued Functions 10 5. Abstract Perturbation Theory and Applications to Birman-Schwinger-Type Operators 12 6. An Index Formula for the Weyl-Titchmarsh Function Associated to Closed Extensions of Dual Pairs 17 References 22where (with the symbol denoting contour integrals)2) for 0 < ε < ε 0 and D(z 0 ; ε 0 )\{z 0 } ⊂ ρ(T ); here D(z 0 ; r 0 ) ⊂ C is the open disk with center z 0 and radius r 0 > 0, and C(z 0 ; r 0 ) = ∂D(z 0 ; r 0 ) the corresponding circle. The geometric multiplicity m g (z 0 ; T ) of an eigenvalue z 0 ∈ σ p (T ) is defined by m g (z 0 ; T ) = dim(ker((T − z 0

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Boundary data maps and Krein's resolvent formula for Sturm-Liouville operators on a finite interval

Cited by 22 publications

References 29 publications

Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

Effective computation of traces, determinants, and ζ-functions for Sturm–Liouville operators

On the index of meromorphic operator-valued functions and some applications

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