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

A quasi-exponential is an entire function of the form eᶜᵘp(u), where $p(u)$ is a polynomial and c ∈ C. Let V = eh₁up₁(u), , ehNupN(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\"{u}cker 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_λG(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_λG(t) is positive semidefinite, which we establish using partial traces. We also show that if h₁ = ⋯ = hN = 0 then T_λG(t) equals β^λ(t), which is the universal Pl\"ucker coordinate for the Wronski map on spaces of polynomials introduced in [KP23].