The Howland-Kato commutator problem. II

We continue the search, begun by T. Kato, for all pairs of real and bounded measurable functions {f,g} on the line that result in a positive commutator i[f(P),g(Q)]. Here P and Q are the usual Heisenberg operators: Q is multiplication by the coordinate x in L2(R,dx) and P=−id/dx on the appropriate domains. We prove a number of partial results including a connection with Loewner's celebrated theorem on matrix monotone functions.