Positivity and Universal Plücker Coordinates for Spaces of Quasi-Exponentials

Abstract Dedicated to Frank Sottile on the occasion of his 60th birthday. 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_{1}u}p_{1}(u), \dots , e^{h_{N}u}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–Plücker 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 [1] 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 [4].