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]:=∑m j,k=0 D jajkD ku=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 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 ∈ [0,m], j + k = 2m, as well as b and the boundary conditions at 0 and at b (if any). Up until quite recently, the most satisfactory results in inverse spectral theory (results essentially due to Borg [4], Marčenko [18] and Gelfand and Levitan [6], and for alternative variants see also [15] and [16]) concerned only the very simples