Positivity and universal Pl\"ucker coordinates for spaces of quasi-exponentials

TL;DR

This paper proves that certain spaces of quasi-exponentials are totally nonnegative when their Wronskian zeros are real, using higher Gaudin Hamiltonians as universal Plücker coordinates.

Contribution

It establishes a connection between higher Gaudin Hamiltonians and Plücker coordinates for spaces of quasi-exponentials, demonstrating total nonnegativity under specific conditions.

Findings

Spaces of quasi-exponentials with real zeros in their Wronskian are totally nonnegative.

Higher Gaudin Hamiltonians serve as universal Plücker coordinates for these spaces.

Total nonnegativity follows from the positive semidefiniteness of these Hamiltonians.

Abstract

A quasi-exponential is an entire function of the form $e^{cu}p(u)$, where $p(u)$ is a polynomial and $c \in \mathbb{C}$. Let $V = \langle e^{h_1u}p_1(u), \dots, e^{h_Nu}p_N(u) \rangle$ be a vector space with a basis of quasi-exponentials. We show that if $h_1, \dots, h_N$ are nonnegative and all of the complex zeros of the Wronskian $\operatorname{Wr}(V)$ are real, then $V$ is totally nonnegative in the sense that all of its Grassmann-Pl\"{u}cker coordinates defined by the Taylor expansion about $u=t$ are nonnegative, for any real $t$ greater than all of the zeros of $\operatorname{Wr}(V)$. Our proof proceeds by showing that the higher Gaudin Hamiltonians $T_\lambda^G(t)$ introduced in [ALTZ14] are universal Pl\"ucker coordinates about $u=t$ for the Wronski map on spaces of quasi-exponentials. The result that $V$ is totally nonnegative follows from the fact that $T_\lambda^G(t)$ is positive semidefinite, which we establish using partial traces. We also show that if $h_1 = \cdots = h_N = 0$ then $T_\lambda^G(t)$ equals $\beta^\lambda(t)$, which is the universal Pl\"ucker coordinate for the Wronski map on spaces of polynomials introduced in [KP23].