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

Abstract

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

On traces and modified Fredholm determinants for half-line Schrödinger operators with purely discrete spectra

“…We refer to Section 3 for an application of (2.8) to half-line Schrödinger operators with potentials diverging at infinity. For additional background and applications of (modified) Fredholm determinants to ordinary differential operators we also refer to [2], [3], [5], [7], [8], [10], [16]- [21], [23], and the extensive literature cited therein.…”

Spectral $$\varvec{\zeta }$$-functions and $$\varvec{\zeta }$$-regularized functional determinants for regular Sturm–Liouville operators

“…In the second part, we discuss Fredholm determinants, traces of resolvents, and spectral ζ -functions associated with these regular Sturm-Liouville problems. For background as well as relevant material in this context we refer to [3,7,12,13,17,19,20,[25][26][27][28][29]31,33,38,39,42,[49][50][51]53,[57][58][59]62,64,71], [72,Sects. 5.4,5.5,6.3], [75].…”

“…Next we state some of the most pertinent concepts and results summarized from [33] (in particular, Section 3) and will then illustrate how this permits one to effectively calculate certain values for the spectral ζ -functions of the regular Sturm-Liouville operators considered.…”

“…The principal motivation for this paper is to illustrate how a recently developed unified approach to the computation of Fredholm determinants, traces of resolvents, and ζfunctions in [33] can be used to efficiently compute certain values of spectral ζ -functions associated with regular Sturm-Liouville operators as well as give the full analytic continuation of the ζ -function through a Liouville transformation and finally provide an explicit expression for the ζ -regularized functional determinant. In Sect.…”

“…In particular, we consider separated boundary conditions when zero is not an eigenvalue, or, when it is (necessarily) a simple eigenvalue, and coupled boundary conditions when either zero is not an eigenvalue, or, an eigenvalue of multiplicity (necessarily) at most two. (For more details in this context see [33] as well as [34,Ch. 3], [76,Sect.…”