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

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 Schrodinger operators on a half-line, are provided and further illustrated with an array of examples. In addition, we consider a class of half-line Schrodinger 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].