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

A quasi-exponential is an entire function of the form e^cup (u), where p (u) is a polynomial and c C. Let V = e^h₁up₁ (u), , e^hNupN (u) be a vector space with a basis of quasi-exponentials. We show that if h₁, , hN are nonnegative and all of the complex zeros of the Wronskian Wr (V) are real, then V is totally nonnegative in the sense that all of its Grassmann-Pl\"ucker coordinates defined by the Taylor expansion about u=t are nonnegative, for any real t greater than all of the zeros of Wr (V). Our proof proceeds by showing that the higher Gaudin Hamiltonians T_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_G (t) is positive semidefinite, which we establish using partial traces. We also show that if h₁ = = hN = 0 then T_G (t) equals ^ (t), which is the universal Pl\"ucker coordinate for the Wronski map on spaces of polynomials introduced in KP23.