Advances in Differential Equations

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

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

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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

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.


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