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

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].