Titchmarsh-Weyl M-function asymptotics and some results in the inverse spectral theory for vector-valued Sturm-Liouville equations and a certain higher order ordinary differential equation

Sammanfattning: This discourse is constituted by two separate reprots, where the first one offers an elementary deduction of the leading order term asymptotics for the Titchmarsh-Weyl M-function corresponding to a vector-valued Sturm-Liouville equation of the form -(PU')'+QU=zu, xin[0,b), with P^{-1},W,Q being hermitean with locally integrable entries; and under some additional conditions on P^{-1} and W. In the special case of P=W=I, we give some further asymptotic results for the same M-function. In this case, we also prove that the corresponding spectral measure determines the equation uniquely up to conjugation by a constant and unitary matrix R, and we finish this presentation by giving a local form of the Borg-Marchenko theorem in the above case (cf. [GS2, Chapter 3.]); a theorem which is due to Simon, [S], in the scalar case. The object of the second report is to study a higher order ordinary differential equation of the form sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}=zu, xin[0,b), where D=id/dx, and where the coefficients a_{jk}, j,kin[0,m], with a_{mm}=1, satisfy certain regularity conditions and are chosen so that the matrix (a_{jk}) is hermitean. We will also assume that m>1. More precisely, we will prove, using Paley-Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. We will also discuss under which additional conditions the spectral measure uniquely determines the coefficients a_{jk}, j,kin[0,m], j+k<2m, as well as b and the boundary conditions at 0 and (if any) at b.