Computing Traces, Determinants, and $$\zeta $$-Functions for Sturm–Liouville Operators: A Survey

Gesztesy, Fritz; Kirsten, Klaus

doi:10.1007/978-3-030-12661-2_7

Cited by 3 publications

(4 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proposed approach exploits in a fundamental way the Taylor and/or Laurent expansion of the resolvent function (or Neumann series) corresponding to linear operators. This approach aligns with the recent work of Gesztesy and Kirsten [23,24,25]. Our procedure explicitly calculates expressions of these operators for all GSARC problems.…”

Section: From the Resolvent To The Integral Operatorsupporting

confidence: 56%

“…The dimension of the range corresponding to P is the algebraic multiplicity, m(T ; 0) of z = 0 [23,24,25]:…”

Section: Review Of Resolvents and Green's Functionsmentioning

confidence: 99%

“…We also investigate a couple of non-GSARC problems that appear in the literature, namely the periodic and anti-periodic cases treated in [22]. Our results in this article focus on perturbation questions not pursued in [22,23,24,25].…”

Section: Our One-dimensional Settingmentioning

confidence: 99%

See 2 more Smart Citations

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

Hermi

Saito

2022

Preprint

View full text Add to dashboard Cite

We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green's function on the real line. This free space Green's function corresponds to the nonlocal boundary value problem proposed earlier by Saito [41]. These are polynomial perturbations of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.

show abstract

Section: From the Resolvent To The Integral Operatorsupporting

confidence: 56%

“…The dimension of the range corresponding to P is the algebraic multiplicity, m(T ; 0) of z = 0 [23,24,25]:…”

Section: Review Of Resolvents and Green's Functionsmentioning

confidence: 99%

See 1 more Smart Citation

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

Hermi

Saito

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Recently, there has been quite some interest in concrete computations of ζ-determinants and Fredholm determinant. We mention, e.g., [GK19a,GK19b,GK19c] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Zeta and Fredholm determinants of self-adjoint operators

Hartmann¹,

Lesch²

2021

Preprint

View full text Add to dashboard Cite

Let L be a self-adjoint invertible operator in a Hilbert space such that L −1 is p-summable. Under a certain discrete dimension spectrum assumption on L, we study the relation between the (regularized) Fredholm determinant, det p(I+z•L −1 ), on the one hand and the zeta regularized determinant, det ζ (L+z), on the other. One of the main results is the formulaWe show that the derivatives d j dz j log det ζ (L + z)| z=0 can be expressed in terms of (regularized) zeta values and heat trace coefficients of L. Furthermore, we give a general criterion in terms of the heat trace coefficients (and which is, e.g., fulfilled for large classes of elliptic operators) which guarantees that the constant term in the asymptotic expansion of the Fredholm determinant, log det p(I+z•L −1 ), equals the zeta determinant of L.

show abstract

Zeta and Fredholm determinants of self-adjoint operators

Hartmann

Lesch

2022

Journal of Functional Analysis

View full text Add to dashboard Cite

Computing Traces, Determinants, and $$\zeta $$-Functions for Sturm–Liouville Operators: A Survey

Cited by 3 publications

References 36 publications

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

Zeta and Fredholm determinants of self-adjoint operators

Zeta and Fredholm determinants of self-adjoint operators

Contact Info

Product

Resources

About