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A spectral Szegő theorem on the real line. (English) Zbl 1428.42046

Summary: We characterize even measures \(\mu = w d x + \mu_s\) on the real line \(\mathbb{R}\) with finite entropy integral \(\int_{\mathbb{R}} \frac{\log w(t)}{1 + t^2} d t > - \infty\) in terms of \(2 \times 2\) Hamiltonians generated by \(\mu\) in the sense of the inverse spectral theory. As a corollary, we obtain criterion for spectral measure of Krein string to have converging logarithmic integral.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34A55 Inverse problems involving ordinary differential equations
81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
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