## Advances in Differential Equations

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

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

D.J. Gilbert, B.J. Harris, and S.M. Riehl

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1369057708

**Mathematical Reviews number (MathSciNet)**

MR3086669

**Zentralblatt MATH identifier**

1280.34083

**Subjects**

Primary: 34B20: Weyl theory and its generalizations 34B24: Sturm-Liouville theory [See also 34Lxx] 47B25: Symmetric and selfadjoint operators (unbounded) 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47)

#### Citation

Gilbert, D.J.; Harris, B.J.; Riehl, S.M. Higher derivatives of spectral functions associated with one-dimensional Schrödinger operators II. Adv. Differential Equations 18 (2013), no. 7/8, 609--632. https://projecteuclid.org/euclid.ade/1369057708