Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T10:43:33.257Z Has data issue: false hasContentIssue false

A UNIQUENESS THEOREM IN THE INVERSE SPECTRAL THEORY OF A CERTAIN HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATION USING PALEY–WIENER METHODS

Published online by Cambridge University Press:  20 July 2005

E. ANDERSSON
Affiliation:
Center for Mathematical Sciences Mathematics, Faculty of Science, University of Lund, Box 118, SE-221 00 Lund, [email protected]
Get access

Abstract

The paper examines a higher-order ordinary differential equation of the form $\mathcal{P}[u]\,{:=} \sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}u \,{=}\, \lambda{u}, x\in[0,b)$, where $D\,{=}\,i({d}/{dx})$, and where the coefficients $a_{jk}$, $j,k\in[0,m]$, with $a_{mm}\,{=}\,1$, satisfy certain regularity conditions and are chosen so that the matrix $(a_{jk})$ is hermitean. It is also assumed that $m\,{>}\,1$. More precisely, it is proved, using Paley–Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. The paper also discusses under which additional conditions the spectral measure uniquely determines the coefficients $a_{jk}$, $j,k\in[0,m]$, $j+k\neq{2m}$, as well as $b$ and the boundary conditions at 0 and at $b$ (if any).

Type
Notes and Papers
Copyright
The London Mathematical Society 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)