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Letters in Mathematical Physics

, Volume 100, Issue 2, pp 151–160 | Cite as

Reality Property of Discrete Wronski Map with Imaginary Step

  • Evgeny Mukhin
  • Vitaly Tarasov
  • Alexander Varchenko
Article

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.

Mathematics Subject Classification (2010)

14P99 17B37 82B23 

Keywords

discrete Wronski map B. and M. Shapiro conjecture Bethe ansatz XXX model 

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Copyright information

© Springer 2011

Authors and Affiliations

  • Evgeny Mukhin
    • 1
  • Vitaly Tarasov
    • 1
    • 2
  • Alexander Varchenko
    • 3
  1. 1.Department of Mathematical SciencesIndiana University—Purdue University IndianapolisIndianapolisUSA
  2. 2.St. Petersburg Branch of Steklov Mathematical InstituteSt. PetersburgRussia
  3. 3.Department of MathematicsUniversity of North Carolina at Chapel HillChapel HillUSA

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