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

Fritz Gesztesy () and Klaus Kirsten ()
Additional contact information
Fritz Gesztesy: Baylor University
Klaus Kirsten: Baylor University

A chapter in Analysis and Operator Theory, 2019, pp 93-148 from Springer

Abstract: Abstract The principal aim of this contribution is to survey an effective and unified approach to the computation of traces of resolvents (and resolvent differences), (modified) Fredholm determinants, $$\zeta $$ -functions, and $$\zeta $$ -function regularized determinants associated with linear operators in a Hilbert space. In particular, we detail the connection between Fredholm and $$\zeta $$ -function regularized determinants. Concrete applications of our formalism to general (i.e., three-coefficient) regular Sturm–Liouville operators on bounded intervals with various (separated and coupled) boundary conditions, and Schrödinger operators on a half-line, are provided and further illustrated with an array of examples. In addition, we consider a class of half-line Schrödinger operators $$(- d^2/dx^2) + q$$ on $$(0,\infty )$$ with purely discrete spectra. Roughly speaking, the class considered is generated by potentials q that, for some fixed $$C_0 > 0$$ , $$\varepsilon > 0$$ , $$x_0 \in (0, \infty )$$ , diverge at infinity of the type $$q(x) \ge C_0 x^{(2/3) + \varepsilon _0}$$ for all $$x \ge x_0$$ . We treat all self-adjoint boundary conditions at the left endpoint 0. This manuscript surveys our recent two papers [19, 20].

Keywords: Traces; (Modified) Fredholm determinants; Semi-separable integral kernels; Sturm–Liouville operators; Discrete spectra; Primary: 47A10; 47B10; 47G10; Secondary: 34B27; 34L40 (search for similar items in EconPapers)
Date: 2019
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-12661-2_7

Ordering information: This item can be ordered from
http://www.springer.com/9783030126612

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

Access Statistics for this chapter

More chapters in Springer Optimization and Its Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().