# Explicit Krein Resolvent Identities for Singular Sturm-Liouville Operators with Applications to Bessel Operators

## TL;DR

This paper develops explicit Krein resolvent identities for singular Sturm-Liouville operators, enabling precise spectral analysis and trace computations for Bessel operators with applications to spectral shift functions.

## Contribution

It introduces explicit Krein resolvent identities for singular Sturm-Liouville operators and applies them to compute spectral traces and shift functions for Bessel operators.

## Key findings

- Derived explicit Krein resolvent identities for singular Sturm-Liouville operators.
- Computed the trace of the resolvent difference for Bessel operators.
- Explicitly determined the spectral shift function for the Bessel operator pair.

## Abstract

We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of the resolvent difference of a pair of self-adjoint realizations of the Bessel expression $-d^2/dx^2+(\nu^2-(1/4))x^{-2}$ on $(0,\infty)$ for values of the parameter $\nu\in[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.