Starting with a time-independent Hamiltonian h and an appropriately chosen solution of the von Neumann equation $i\dot{\rho }\mathrm{}\left(t\right)=[h,\rho \mathrm{}(t\left)\mathrm{}\right]$ we construct its binary-Darboux partner $h_1\mathrm{}\left(t\right)\mathrm{}$ and an exact scattering solution of $i{\dot{\rho }}_1\mathrm{}\left(t\right)=\left[h_1\mathrm{}\right(t),{\rho }_1\mathrm{}(t\left)\mathrm{}\right],$ where $h_1\mathrm{}\left(t\right)\mathrm{}$ is time dependent and not isospectral to h. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where h corresponds to a one-dimensional harmonic oscillator. The resulting $h_1\mathrm{}\left(t\right)\mathrm{}$ represents a scattering of a solitonlike pulse on a three-level system.

• Received 16 July 1999

DOI:https://doi.org/10.1103/PhysRevE.61.3325