Inverse spectral theory for Sturm-Liouville operators with distributional potentials (Q2869831)

The authors study the inverse spectral theory of very general singular second order differential operators generated by the expression, for \(a<x<b,\) by NEWLINE\[NEWLINE \tau\left( y\right) (x):=\frac{1}{r(x)}\left( -p(x)\left[ y^{\prime}(x)+s(x)y(x)\right] ^{\prime}\right) +s(x)p(x)\left[ y^{\prime}(x)+s(x)y(x)\right] +q(x)y(x) NEWLINE\]NEWLINE where it is assumed that \(p(x)\neq0,\;r(x)>0\) almost everywhere and \(\;p^{-1},q,r,s\in L^{1,\mathrm{loc}}\left( a,b\right) \). Note that quasi-derivatives and distributional coefficients are also allowed in the expression of \(\tau.\) These operators cover the standard Sturm-Liouville operators where \(p(x)=r(x)=1\) and \(s(x)=0\), impedance form, and also Krein types of strings when \(p(x)=1\) and \(s(x)=q(x)=0\), and thus their inverse spectral theory unifies both the Gelfand-Levitan theory which is based on transformation operators and also M.G.Krein's string inverse spectral theory which relies on function theory and DeBranges spaces. Most of the uniqueness proofs make use of the asymptotics and representation of the Weyl-Titchmarsh-Kodaira function as a Nevanlinna-Herglotz function.