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Journal of the London Mathematical Society 2005 72(1):169-184; doi:10.1112/S0024610704005770
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© The London Mathematical Society

A Uniqueness Theorem in the Inverse Spectral Theory of a Certain Higher-Order Ordinary Differential Equation Using Paley–Wiener Methods

E. Andersson

Center for Mathematical Sciences Mathematics, Faculty of Science, University of Lund Box 118, SE-221 00 Lund, Sweden erik.andersson{at}math.lu.se

Received 19 September 2003. Revision received 19 February 2005.

The paper examines a higher-order ordinary differential equation of the form Formula where D = i(d/dx), and where the coefficients ajk, j,k isin [0,m], with amm = 1, satisfy certain regularity conditions and are chosen so that the matrix (ajk) 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 ajk, j,k isin [0,m], j + k != 2m, as well as b and the boundary conditions at 0 and at b (if any).


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