the quantum trajectory evolution equations, i.e. the continuity equation ensures that the probability \[i.e. density, $R(x,t)^2$, times volume element\] carried by individual quantum trajectories is conserved over the course of their time evolution.

### CPWM decomposition for stationary states {#bipolar}

In paper I, we derived a unique bipolar decomposition, (x) = \_+(x) + \_-(x), \[bipolardecomp\] for stationary eigenstates $\Psi(x)$ of 1D Hamiltonians of the form, such that:

1.  [$\Ppm(x)$ are themselves (non-$L^2$) solutions to the  equation, with the same eigenvalue, $E$, as $\Psi(x)$ itself.]{}

2.  [The invariant flux values, $\pm F$, of the two solutions, $\Ppm(x)$, equal those of the two semiclassical (WKB) solutions.]{}

3.  [The median of the enclosed action, $x_0$, equals that of the semiclassical solutions.]{}

There are other important properties of the $\Ppm(x)$,[@poirier04bohmI] as discussed in Sec. \[intro\], and in [Ref. ]{}. Nevertheless, the above three conditions are sufficient to uniquely specify the decomposition. In the special case of bound (i.e. $L^2$) stationary states, the real-valuedness of $\Psi(x)$ implies that the $\Ppm(x)$ are complex conjugates of each other.

Scattering systems {#scattering}
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It is natural to ask to what extent the above analysis may be generalized for scattering potentials. Certainly, $\Psi(x)$ itself is no longer $L^2$, nor 