Optimality of the free quantum evolution: the general case with nodes

Extending and refining a preceding work, the free evolution of a quantum wave function, with fixed initial and final modulus, is considered in the general case where nodes are allowed. The method is based on considering a suitable simple functional related to the quadratic optimal transport cost in discrete time. Under reasonable assumptions it is shown that, if nodes are present, the free evolution of the wave function with fixed initial and final modulus is a sort of local minimum. If no nodes are present the minimum is global, suggesting intrinsic instability of the nodes in a free quantum evolution.