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A Uniqueness Theorem in the Inverse Spectral Theory of a Certain Higher‐Order Ordinary Differential Equation Using Paley–Wiener Methods
Author(s) -
Andersson E.
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704005770
Subject(s) - mathematics , ordinary differential equation , uniqueness , measure (data warehouse) , order (exchange) , mathematical analysis , uniqueness theorem for poisson's equation , inverse , function (biology) , spectral measure , matrix (chemical analysis) , differential equation , chemistry , geometry , finance , chromatography , database , evolutionary biology , computer science , economics , biology
The paper examines a higher‐order ordinary differential equation of the form P [ u ] : = ∑ j , k = 0 mD ja j kD k u = λ u , x ∈ [ 0 , b ) where D = i ( d / dx ), and where the coefficients a jk , j,k ∈ [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 ∈ [0, m ], j + k ≠ 2 m , as well as b and the boundary conditions at 0 and at b (if any).