An inverse spectral theorem for Kreĭn strings with a negative eigenvalue

摘要

A string is a pair ${(L, mathfrak{m})}$ where ${L in[0, infty]}$ and ${mathfrak{m}}$ is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of ${mathfrak{m}}$ as its mass density. To each string a differential operator acting in the space ${L^2(mathfrak{m})}$ is associated. Namely, the Kreĭn–Feller differential operator ${-D_{mathfrak{m}}D_x}$ ; its eigenvalue equation can be written, e.g., as $$f^{prime}(x) + z int_0^L f(y),dmathfrak{m}(y) = 0,quad x inmathbb R, f^{prime}(0-) = 0.$$ A positive Borel measure τ on ${mathbb R}$ is called a (canonical) spectral measure of the string ${textsc S[L, mathfrak{m}]}$ , if there exists an appropriately normalized Fourier transform of ${L^2(mathfrak{m})}$ onto L 2(τ). In order that a given positive Borel measure τ is a spectral measure of some string, it is necessary that: (1) ${int_{mathbb R} frac{dtau(lambda)}{1+|lambda|} infty}$ . (2) Either ${{rm supp} tau subseteq [0, infty)}$ , or τ is discrete and has exactly one point mass in (−∞, 0). It is a deep result, going back to Kreĭn in the 1950’s, that each measure with ${int_{mathbb R}frac{dtau(lambda)}{1+|lambda|} infty}$ and ${{rm supp} tau subseteq [0, infty)}$ is a spectral measure of some string, and that this string is uniquely determined by τ. The question remained open, which conditions characterize whether a measure τ with ${{rm supp} tau notsubseteq [0, infty)}$ is a spectral measure of some string. In the present paper, we answer this question. Interestingly, the solution is much more involved than the first guess might suggest.