Note on a diffraction amplification problem

We investigate the solution of the equation \partial_t{\cal E}(x,t)-i{\cal D}\partial_x^2 {\cal E}(x,t)= \lambda\vert S(x,t)\vert^2{\cal E}(x,t) , for x in a circle and S(x, t) a Gaussian stochastic field with a covariance of a particular form. It is shown that the coupling lgr_c at which \langle\vert {\cal E}\vert\rangle diverges for t geq 1 (in suitable units), is always less or equal for {\cal D}\gt0 than {\cal D}=0 .