Abstract
We study several related aspects of reflectionless Jacobi matrices. First, we discuss the singular part of the corresponding spectral measures. We then show how to identify sets on which measures are reflectionless by looking at the logarithmic potentials of these measures.
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Bruckner, A.M.: Differentiation of Real Functions. Lecture Notes in Mathematics 659, Berlin: Springer, 1978
Cima, J.A., Matheson, A.L., Ross, W.T.: The Cauchy Transform. Mathematical Surveys and Monographs 125, Providence, RI: Amer. Math. Soc., 2006
Dombrowski J.: Quasitriangular matrices. Proc. Amer. Math. Soc. 69, 95–96 (1978)
Gesztesy F., Krishna M., Teschl G.: On isospectral sets of Jacobi operators. Commun. Math. Phys. 181, 631–645 (1996)
Gesztesy F., Yuditskii P.: Spectral properties of a class of reflectionless Schrödinger operators. J. Funct. Anal. 241, 486–527 (2006)
Gesztesy, F., Zinchenko, M.: Local spectral properties of reflectionless Jacobi, CMV, and Schrödinger operators. http://arxiv.org/abs/0803.3177 [math.sp], 2008
Jaksic V., Last Y.: A new proof of Poltoratski’s theorem. J. Funct. Anal. 215, 103–110 (2004)
Martin, M., Putinar, M.: Lectures on Hyponormal Operators. Operator Theory: Advances and Applications 39, Basel: Birkhäuser Verlag, 1989
Melnikov, M., Poltoratski, A., Volberg, A.: Uniqueness theorems for Cauchy integrals. http://arxiv.org/abs/0704.0621 [math.sp], 2007
Nazarov, F., Volberg, A., Yuditskii, P.: Reflectionless measures with a point mass and singular continuous component. http://arxiv.org/abs/0711.0948 [math.sp], 2008
Poltoratski A.: Boundary behavior of pseudocontinuable functions. St. Petersburg Math. J. 5, 389–406 (1994)
Ransford, T.: Potential Theory in the Complex Plane, London Mathematical Society Student Texts 28, Cambridge: Cambridge University Press, 1995
Remling, C.: The absolutely continuous spectrum of Jacobi matrices. http://arxiv.org/abs/0706.1101 [math.sp], 2007
Saks, S.: Theory of the Integral. Second revised edition, New York: Dover Publications, 1964
Simon B.: Equilibrium measures and capacities in spectral theory. Inverse Probl. Imaging 1, 713–772 (2007)
Simon, B.: Weak convergence of CD kernels and applications. To appear in Duke Math. J.
Simon B., Spencer T.: Trace class perturbations and the absence of absolutely continuous spectra. Commun. Math. Phys. 125, 113–125 (1989)
Sodin M., Yuditskii P.: Almost periodic Jacobi matrices with homogeneous spectrum, infinite- dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7, 387–435 (1997)
Stahl, H., Totik, V.: General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications 43, Cambridge: Cambridge University Press, 1992
Thomson, B.S.: Real Functions, Lecture Notes in Mathematics 1170, Berlin: Springer-Verlag, 1985
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Communicated by B. Simon
A. P.’s work is supported in part by NSF grant DMS 0800300.
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Poltoratski, A., Remling, C. Reflectionless Herglotz Functions and Jacobi Matrices. Commun. Math. Phys. 288, 1007–1021 (2009). https://doi.org/10.1007/s00220-008-0696-x
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DOI: https://doi.org/10.1007/s00220-008-0696-x