Positivity and Universal Plücker Coordinates for Spaces of Quasi-Exponentials

Dedicated to Frank Sottile on the occasion of his 60th birthday. A quasi-exponential is an entire function of the form eᶜᵘp(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–Plücker 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_λ 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_λ G(t) is positive semidefinite, which we establish using partial traces. We also show that if h₁ = ⋯ = hN = 0 then T_λ G(t) equals β ^λ (t), which is the universal Plücker coordinate for the Wronski map on spaces of polynomials introduced in [4].