Krein\u27s identity and trace formulas for half-line Schrodinger operators

We consider self-adjoint extensions of the minimal operator generated by the differential expression $\cL = - d^2/dx^2+V$ on the half-line $[0,\infty)$, where $V$ is a real-valued function integrable with respect to the weight $1+x$. The self-adjoint extensions are of Schr\ odinger-type and form a one-parameter family formally given by $H_{\alpha}=-d^2/dx^2+V$, $\alpha\in[0,\pi)$, with the boundary condition $\sin(\alpha)f\u27(0) = \cos(\alpha)f(0)$ at $x=0$. We derive a formula that relates the resolvent operator of $H_{\alpha}$ to the resolvent operator of $H_0$ in terms of the Jost solution corresponding to the underlying differential equation $\mathcal{L}u=zu$. Combining this resolvent formula and properties of the Jost solution, we compute the trace of the difference of the resolvents of $H_{\alpha}$ and the free operator $H_{\alpha}^{(0)}$ with $V\equiv 0$ in terms of the parameter $\alpha$ and the Jost function for $\cL u=zu$