Higher order trace relations for Schrödinger operators

We extend the trace formula recently proven for general one-dimensional Schrödinger operators which obtains the potential V(x) from a function ξ(x, λ) by deriving trace relations computing moments of ξ(λ, x) dλ in terms of polynomials in the derivatives of V at x. We describe the relation of those polynomials to KdV invariants. We also discuss trace formulae for analogs of ξ associated with boundary conditions other than the Dirichlet boundary condition underlying ξ