Higher derivatives of spectral functions associated with one-dimensional Schrödinger operators II

Abstract

We consider the spectral function, $\rho_{\alpha}(\lambda)$, associated with the linear, second-order differential equation $$ y{''} + (\lambda - q) y = 0 \quad \mbox{on} \ [0,\infty) $$ with the initial condition $$ y(0) \cos (\alpha) + y'(0) \sin (\alpha) = 0 \quad \mbox{for some} \ \ \alpha \in [0,\pi). $$ It is shown that if $(1+x)^n q(x) \in L^1[0,\infty)$, then $(n+1)$ derivatives of $\rho_0(\lambda)$ exist and are continuous for $\lambda>0$. Under similar conditions, the derivatives are explicitly computed for $\lambda \ge \Lambda_0$ where $\Lambda_0$ is computable.