Explicit formulae for the powers of a Schrödinger-like ordinary differential operator II

We derive some new formulae in the Schur pseudodifferential operator calculus with respect to one real variable x, namely for (i) the inverse of a Schrödinger-like operator Dr+u, where D=d/dx, r is a positive integer, and u=u(x) is a smooth function; (ii) arbitrary integer powers of D+u; (iii) the square root of the Schrödinger operator D2+u. The result (iii) is closely connected with the theory of the KdV hierarchy.