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Reality Property of Discrete Wronski Map with Imaginary Step

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Abstract

For a set of quasi-exponentials with real exponents, we consider the discrete Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We prove that if the coefficients of the discrete Wronskian are real and the imaginary parts of its roots are bounded by |h|, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result is a generalization of the statement of the B. and M. Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe ansatz for the XXX model.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indiana University—Purdue University Indianapolis, 402 North Blackford St., Indianapolis, IN, 46202-3216, USA

    Evgeny Mukhin & Vitaly Tarasov

  2. St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia

    Vitaly Tarasov

  3. Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599-3250, USA

    Alexander Varchenko

Authors
  1. Evgeny Mukhin
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  2. Vitaly Tarasov
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  3. Alexander Varchenko
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Correspondence to Evgeny Mukhin.

Additional information

E. Mukhin—Supported in part by NSF grant DMS-0900984.

V. Tarasov—Supported in part by NSF grant DMS-0901616.

A. Varchenko—Supported in part by NSF grant DMS-0555327.

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Mukhin, E., Tarasov, V. & Varchenko, A. Reality Property of Discrete Wronski Map with Imaginary Step. Lett Math Phys 100, 151–160 (2012). https://doi.org/10.1007/s11005-011-0521-x

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  • DOI: https://doi.org/10.1007/s11005-011-0521-x

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