TIME-TO-ENERGY TRANSFORM OF WAVEPACKETS USING ABSORBING POTENTIALS - TIME-INDEPENDENT WAVEPACKET-SCHRODINGER AND WAVEPACKET-LIPPMANN-SCHWINGER EQUATIONS

It is shown that one may use an L2 basis, matrix representation of the Hamiltonian, including a negative imaginary absorbing potential, to carry out arbitrarily long-time evolution of wavepackets. The time-to-energy Fourier transform of the wavepacket is carried out analytically, yielding a new type of time-independent scattering equation in which the ''source'' of scattered waves is the initial (t = 0) L2 wavepacket used in the time-dependent propagation. Alternatively, one obtains the analogous time-independent, inhomogeneous wavepacket-Schrodinger equation. A banded representation of the Hamiltonian is achieved by the use of distributed approximating function theory to evaluate the kinetic energy. The resulting new time-independent wavepacket equations are solved both by matrix diagonalization of the Hamiltonian, and as inhomogeneous linear algebraic equations. The approach is illustrated by application to electron scattering (in one dimension) by a double barrier potential.