Positivity and universal Plücker coordinates for spaces of quasi-exponentials

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 $V=⟨e^{h_{1}u}{p}_1(u),\dots ,e^{h_{N}u}{p}_N(u)⟩$ 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 $\mathrm{Wr}(V)$ are real, then $V$ is totally nonnegative in the sense that all of its Grassmann-Plücker coordinates defined by the Taylor expansion about $u=t$ are nonnegative, for any real $t$ greater than all of the zeros of $\mathrm{Wr}(V)$. Our proof proceeds by showing that the higher Gaudin Hamiltonians $T_{\lambda }^G(t)$ introduced in [ALTZ14] are universal Plücker 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 Plücker coordinate for the Wronski map on spaces of polynomials introduced in [KP23].

24 pages. v2: Final version