A UNIQUENESS THEOREM IN THE INVERSE SPECTRAL THEORY OF A CERTAIN HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATION

The paper examines a higher-order ordinary differential equation of the form P[u] := m j,k=0 DjajkDku=λu, x ∈ [0, b), where D=i(d/dx), and where the coefficients ajk, j,k ∈ [0,m], with amm =1, satisfy certain regularity conditions and are chosen so that the matrix (ajk) is hermitean. It is also assumed thatm>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 ∈ [0,m], j + k = 2m, as well as b and the boundary conditions at 0 and at b (if any).