On a form of nonlinear dissipative wave mechanics valid in position- and momentum-space

In wave mechanics an appropriate description of a system under the influence of a linearly velocity dependent frictional force can be given by a nonlinear Schroedinger equation (NLSE) with logarithmic nonlinearity. However, the particular logarithmic form of the dissipative nonlinear frictional term in the NLSE is connected with the definition of the momentum- or velocity-operator in position-space. Therefore, in momentum-space, this form of the NLSE is no longer correct to describe the same physical situation. This can be seen, e.g., from the fact that, in contrast to the linear case, the Fourier transform of the solution of the NLSE in position-space does not fulfill anymore the logarithmic NLSE in momentum-space. It will be shown, using results obtained from the theory in position-space, that it is possible to find a form of the nonlinear dissipative frictional term which is valid in position-as well as in momentum-space. Using this form, the NLSE looks like a diffusion equation with complex diffusion coefficient, i.e., a combination of a diffusion and a Schroedinger equation. The solution of this NLSE in momentum-space will be discussed