  The Complex Langevin (CL) method to simulate `complex probabilities', ideally
produces expectation values for the observables that converge to a limit equal
to the expectation values obtained with the original complex `probability'
measure. The situation may be spoiled in two ways: failure to converge and
convergence to the wrong limit. It was found long ago that `wrong convergence'
is caused by boundary terms; non-convergence may arise from bad spectral
properties of the various evolution operators related to the CL process. Here
we propose a class of criteria which allow to rule out boundary terms and at
the same time bad spectrum. Ruling out boundary terms in the equilibrium
distribution arising from a CL simulation implies that the so-called
convergence conditions are fulfilled. This in turn has been shown to guarantee
that the expectation values of holomorphic observables are given by complex
linear combinations of $\exp(-S)$ over various integration cycles. If the
spectrum is pathological, however, the CL simulation in general does not
reproduce the integral over the desired real cycle.
