Krein's strings with singular left boundary (Q1002133)

Let \(m\:\mathbb R\to[0,+\infty]\) be a nondecreasing right-continuous function with \(m(-\infty)=0\) (such functions will be called strings). Set \(l=\sup\{x>-\infty\mid m(x)<+\infty\}\) and assume that \(\int_{-\infty}^c|x|\,dm(x)<\infty\) for some \(c<l\). Denote \(M(x)=\int_{-\infty}^xm(y)\,dy\). It is known that there exists a nonnegative measure \(\sigma\) on \([0,+\infty)\) satisfying \(\|f\|^2=\int_0^{\infty}|f(\xi)|^2\,\sigma(d\xi)\) for \(f\) belonging to a certain function space. This measure \(\sigma\) is called a spectral measure for the string~\(m\). The author proves that for an integer \(n\geq1\) the spectral measure \(\sigma\) satisfies the condition \(\int_0^{\infty}\sigma(d\xi)/(\xi^n+1)<\infty\) if and only if \(m\) fulfills \(\int_{-\infty}^cM(x)^{n-1}\,dx<\infty\) with some \(c<l\). Thus, a classical theorem of Krein establishing a one-to-one correspondence between strings with regular left boundary and their spectral measures is extended to a correspondence between certain class of strings with singular left boundary and spectral measures that grow polynomially. In addition, strings with left boundary of limit circle type are treated.