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.

Article information

Source
Adv. Differential Equations, Volume 18, Number 7/8 (2013), 609-632.

Dates
First available in Project Euclid: 20 May 2013