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Direct Cauchy theorem and Fourier integral in Widom domains

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Abstract

We derive the Fourier integral associated with the complex Martin function in the Denjoy domain of the Widom type with the Direct Cauchy Theorem (DCT). As an application we study canonical systems and corresponding transfer matrices generated by reflectionless Weyl-Titchmarsh functions in such domains. The DCT property appears to be crucial in many aspects of the underlying theory.

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References

  1. V.M. Adamjan, D.Z. Arovand M.G. Krein, Infinite Hankel block matrices and related problems of extension, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6 (1971), 87–112.

    MathSciNet  Google Scholar 

  2. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Olivery & Boyd, New York 1965.

    MATH  Google Scholar 

  3. D. Z. Arov and L. Z. Grossman, Scattering matrices in the theory of extensions of isometric operators, Dokl. Akad. Nauk SSSR 270 (1983), 17–20.

    MathSciNet  Google Scholar 

  4. L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, Boston, MA, 1968.

    MATH  Google Scholar 

  5. S. Breimesser and D. Pearson, Asymptotic value distribution for solutions of the Schrodinger equation, Math. Phys. Anal. Geom. 3 (2000), 385–403.

    Article  MathSciNet  Google Scholar 

  6. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

    MATH  Google Scholar 

  7. B. Eichinger, T. VandenBoom and P. Yuditskii, KdV hierarchy via abelian coverings and operator identities, Trans. Amer. Math. Soc. 6 (2019), 1–44.

    Article  MathSciNet  Google Scholar 

  8. A. Eremenko and P. Yuditskii, Comb functions, in Recent Advances in Orthogonal Polynomials, Special Functions, and their Applications, American Mathematical Society, Providence, RI, 2012, pp. 99–11.

    MATH  Google Scholar 

  9. J.B. Garnett, Bounded Analytic Functions, Springer, New York, 2007.

    MATH  Google Scholar 

  10. J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge University Press, Cambridge, 2005.

    Book  Google Scholar 

  11. F. Gesztesy and P. Yuditskii, Spectral properties of a class of reflectionless Schrodinger operators, J. Funct. Anal. 241 (2006), 486–527.

    Article  MathSciNet  Google Scholar 

  12. M. Hasumi, Hardy Classes on Infinitely Connected Riemann Surfaces, Springer, Berlin, 1983.

    Book  Google Scholar 

  13. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I, Springer, Berlin-New York, 1979

    Book  Google Scholar 

  14. A. Kheifets and P. Yuditskii, An analysis and extension of V.P. Potapov’s approach to interpolation problems with applications to the generalized bi-tangential Schur-Nevanlinna-Pick problem and J-inner-outer factorization, in Matrix and Operator Valued Functions, Birkhäuser, Basel, 1994, pp. 133–161.

    Chapter  Google Scholar 

  15. P. Koosis, The Logarithmic Integral. I, Cambridge University Press, Cambridge, 1998.

    MATH  Google Scholar 

  16. M. G. Krein and A. A. Nudel’man, The Markov Moment Problem and Extremal Problems. Ideas and Problems of P. L. Čebyšev and A. A. Markov and their Further Development, American Mathematical Society, Providence, RI, 1977.

    Book  Google Scholar 

  17. P. Lax and R. Phillips, Scattering Theory, Rocky Mountain J. Math. 1 (1971), 173–224.

    Article  MathSciNet  Google Scholar 

  18. V. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.

    Book  Google Scholar 

  19. D. Mumford, Tata Lectures on Theta. II, Birkhauser Boston, Boston, MA, 2007.

    Book  Google Scholar 

  20. B. Sz.-Nagy, C. Foias, H. Bercovici and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Springer, New York, 2010.

    Book  Google Scholar 

  21. N. K. Nikolskii, Treatise on the Shift Operator, Spriger, Berlin, 1986.

    Book  Google Scholar 

  22. L. Pastur and A. Figotin, Spectra of Random and Almost-periodic Operators, Springer, Berlin, 1992.

    Book  Google Scholar 

  23. A. Poltoratski and C. Remling, Reflectionless Herglotz functions and Jacobi matrices, Comm. Math. Phys. 288 (2009), 1007–1021.

    Article  MathSciNet  Google Scholar 

  24. C. Remling, The absolutely continuous spectrum of Jacobi matrices, Ann. of Math. (2) 174 (2011), 125–171.

    Article  MathSciNet  Google Scholar 

  25. C. Remling, Spectral Theory of Canonical Systems, Walter De Gruyter, Berlin, 2018.

    Book  Google Scholar 

  26. R. Romanov, Canonical systems and de Branges spaces, preprint, arXiv:1408.6022 [math.SP]

  27. M. Sodin and P. Yuditskii, Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum, Comment. Math. Helv. 70 (1995), 639–658.

    Article  MathSciNet  Google Scholar 

  28. M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), 387–435.

    Article  MathSciNet  Google Scholar 

  29. A. Volberg and P. Yuditskii, Kotani-Last problem and Hardy spaces on surfaces of Widom type, Invent. Math. 197 (2014), 683–740.

    Article  MathSciNet  Google Scholar 

  30. A. Volberg and P. Yuditskii, Mean type of functions of bounded characteristic and Martin functions in Denjoy domains, Adv.Math. 290 (2016), 860–887.

    Article  MathSciNet  Google Scholar 

  31. H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), 220–269.

    Article  MathSciNet  Google Scholar 

  32. H. Widom, The maximum principle for multiple valued analytic functions, Acta Math. 126 (1971), 63–81.

    Article  MathSciNet  Google Scholar 

  33. P. Yuditskii, On the direct Cauchy theorem in Widom domains: positive and negative examples, Comput. Methods Funct. Theory 11 (2011), 395–414.

    Article  MathSciNet  Google Scholar 

  34. P. Yuditskii, On L1extremal problem for entire functions, J. Approx. Theory 179 (2014), 63–93.

    Article  MathSciNet  Google Scholar 

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Acknowledgment

The author would like to express his gratitude to Benjamin Eichinger and Roman Bessonov for very helpful discussions. This work was supported by the Austrian Science Fund FWF, project no: P29363-N32.

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Correspondence to Peter Yuditskii.

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In the study of mathematics, there is a grave injustice: we put in so much effort, but we get such miserable results…

Larry Zalcman (from a private conversation)

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Yuditskii, P. Direct Cauchy theorem and Fourier integral in Widom domains. JAMA 141, 411–439 (2020). https://doi.org/10.1007/s11854-020-0122-7

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  • DOI: https://doi.org/10.1007/s11854-020-0122-7

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