Reality property of discrete Wronski map with imaginary step

TL;DR

This paper proves that under certain conditions, the complex span of quasi-exponentials with real exponents has a basis of real-coefficient quasi-exponentials, extending the B. and M. Shapiro conjecture to discrete Wronskians.

Contribution

It generalizes the B. and M. Shapiro conjecture to the setting of discrete Wronskians with imaginary step using Bethe ansatz methods.

Findings

Discrete Wronskian roots have imaginary parts at most |h|.

The complex span admits a basis of real-coefficient quasi-exponentials.

Extension of Shapiro conjecture to discrete Wronskians with imaginary step.

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 for every its roots the imaginary part is at most |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.