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

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 $-d^2/dx^2+( u^2-(1/4))x^{-2}$ on $(0,infty)$ for values of the parameter $ uin[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.