Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringSL,m] with a (non-decreasing) mass distribution functionm on some interval 0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on 0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832