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

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).