Direct and inverse spectral problems for generalized strings

Let the function Q be holomorphic in the upper half plane C+ and such that Im Q(z) greater than or equal to 0 and Im zQ(z) greater than or equal to 0 if z is an element of C+. A basic result of M.G. Krein states that these functions Q are the principal Titchmarsh-Weyl coefficients of a (regular or singular) string S[L,m] with a (non-decreasing) mass distribution function m on some interval [0, L] with a left endpoint 0. This string corresponds to the eigenvalue problem df + lambda fdm = 0; f'(0-) = 0. In this note we show that the set of functions Q which are holomorphic in C+ and such that the kernel Q(z)-<(Q(zeta))over bar>/z - <(zeta)over bar> has kappa negative squares on C+ and Im zQ(z) greater than or equal to 0 if z is an element of C+ is the principal Titchmarsh-Weyl coefficient of a generalized. string, which is described by the eigenvalue problem df' t lambda fdm + lambda(2)fdD = 0 on [0, L), f'(0-) = 0. Here kappa is the number of points x where D increases or 0 > m(x + 0) - m(x - 0) greater than or equal to -infinity outside of these points x the function m is locally non-decreasing and the function D is constant.