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Dirac Operators with Exponentially Decaying Entropy

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Abstract

We prove that the Weyl function of the one-dimensional Dirac operator on the half-line \({\mathbb {R}}_+\) with exponentially decaying entropy extends meromorphically into the horizontal strip \(\{0\geqslant \mathop {\textrm{Im}}\nolimits z > -\delta \}\) for some \(\delta > 0\) depending on the rate of decay. If the entropy decreases very rapidly then the corresponding Weyl function turns out to be meromorphic in the whole complex plane. In this situation we show that poles of the Weyl function (scattering resonances) uniquely determine the operator.

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References

  1. Bessonov, R.: Szegő condition and scattering for one-dimensional Dirac operators. Constr. Approx. 51(2), 273–302 (2020)

    Article  MathSciNet  Google Scholar 

  2. Bessonov, R., Denisov, S.: De Branges canonical systems with finite logarithmic integral. Anal. PDE 14(5), 1509–1556 (2021)

    Article  MathSciNet  Google Scholar 

  3. Bessonov, R., Denisov, S.: Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials. J. Funct. Anal. 280(12), 38, 109002 (2021)

  4. Bessonov, R., Denisov, S.: Sobolev norms of \({L}^2\)-solutions to NLS. arXiv:2211.07051 (2022)

  5. Bessonov, R., Denisov, S.: Szegő condition, scattering, and vibration of Krein strings. Invent. Math. 234(1), 291–373 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  6. Damanik, D., Simon, B.: Jost functions and Jost solutions for Jacobi matrices. II. Decay and analyticity. Int. Math. Res. Not., pages Art. ID 19396, 32 (2006)

  7. Denisov, S.: Continuous analogs of polynomials orthogonal on the unit circle and Kreĭn systems. IMRS Int. Math. Res. Surv., pages Art. ID 54517, 148 (2006)

  8. Dyatlov, S., Zworski, M.: Mathematical Theory of Scattering Resonances. Graduate Studies in Mathematics, vol. 200. American Mathematical Society, Providence, RI (2019)

    Google Scholar 

  9. Fried, H.: Green’s Functions and Ordered Exponentials. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  10. Froese, R.: Asymptotic distribution of resonances in one dimension. J. Differ. Equ. 137(2), 251–272 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  11. Garnett, J.: Bounded Analytic Functions. Pure and Applied Mathematics, vol. 96. Academic Press, New York (1981)

    Google Scholar 

  12. Iantchenko, A., Korotyaev, E.: Resonances for 1D massless Dirac operators. J. Differ. Equ. 256(8), 3038–3066 (2014)

    Article  MathSciNet  Google Scholar 

  13. Iantchenko, A., Korotyaev, E.: Resonances for Dirac operators on the half-line. J. Math. Anal. Appl. 420(1), 279–313 (2014)

    Article  MathSciNet  Google Scholar 

  14. Klein, M.: On the absence of resonances for Schrödinger operators with nontrapping potentials in the classical limit. Commun. Math. Phys. 106(3), 485–494 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  15. Korey, M.: Ideal weights: asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation. J. Fourier Anal. Appl. 4(4–5), 491–519 (1998)

    Article  MathSciNet  Google Scholar 

  16. Korotyaev, E., Mokeev, D.: Inverse resonance scattering for Dirac operators on the half-line. Anal. Math. Phys., 11(1):Paper No. 32, 26 (2021)

  17. Krein, M.: Continuous Analogues of Propositions on Polynomials Orthogonal on the Unit Circle. Dokl. Akad. Nauk SSSR (N.S.), 105:637–640 (1955)

  18. Levin, B.: Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150. American Mathematical Society, Providence, RI (1996)

  19. Levitan, B., Sargsjan, I.: Sturm–Liouville and Dirac operators, volume 59 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1991). Translated from the Russian

  20. Matveev, V., Skriganov, M.: Wave operators for a Schrödinger equation with rapidly oscillating potential. Dokl. Akad. Nauk SSSR 202, 755–757 (1972)

    MathSciNet  Google Scholar 

  21. Nevai, P., Totik, V.: Orthogonal polynomials and their zeros. Acta Sci. Math. (Szeged) 53(1–2), 99–104 (1989)

    MathSciNet  Google Scholar 

  22. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Academic Press, New York-London (1979). Scattering theory

  23. Remling, C.: Spectral Theory of Canonical Systems. De Gruyter, Berlin (2018)

    Book  Google Scholar 

  24. Romanov, R.: Canonical systems and de Branges spaces. arXiv:1408.6022 (2014)

  25. Sasaki, I.: Schrödinger operators with rapidly oscillating potentials. Integral Equ. Oper. Theory 58(4), 563–571 (2007)

    Article  Google Scholar 

  26. Simon, B.: Resonances in one dimension and Fredholm determinants. J. Funct. Anal. 178(2), 396–420 (2000)

    Article  MathSciNet  Google Scholar 

  27. Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1, volume 54 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (2005). Classical Theory

  28. Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 2, volume 54 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (2005). Spectral theory

  29. Sjöstrand, J.: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60(1), 1–57 (1990)

    Article  MathSciNet  Google Scholar 

  30. Skriganov, M.: The spectrum of a Schrödinger operator with rapidly oscillating potential. Trudy Mat. Inst. Steklov., 125:187–195, 235 (1973). Boundary value problems of mathematical physics, 8

  31. Szegő, G.: Orthogonal Polynomials, vol. XXIII, 4th edn. American Mathematical Society, Providence, R.I. (1975)

    Google Scholar 

  32. Teplyaev, A.: A note on the theorems of M. G. Krein and L. A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle. J. Funct. Anal. 226(2), 257–280 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I am grateful to Roman Bessonov for numerous discussions and constant attention to this work.

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Authors and Affiliations

  1. St. Petersburg State University, Universitetskaya Nab. 7-9, St. Petersburg, Russia, 199034

    Pavel Gubkin

  2. St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia, 191023

    Pavel Gubkin

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  1. Pavel Gubkin
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Correspondence to Pavel Gubkin.

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Communicated by Sergey Denisov.

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Gubkin, P. Dirac Operators with Exponentially Decaying Entropy. Constr Approx (2024). https://doi.org/10.1007/s00365-024-09678-0

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