Scattering calculations in the oscillator representation: improved convergence and absorbing boundary conditions

The Schrödinger equation is solved for scattering solutions in a hybrid representation for one and twodimensional problems. The solutions are expanded in the eigenstates of the harmonic oscillator in the interaction region and a finite difference grid describes the solution in the near– and far–field. The finite– difference grid has an absorbing boundary layer based on exterior complex scaling. The two representations are coupled through a high–order asymptotic formula. It takes into account the function values and the third derivative in the classical turning points of the oscillator states in coordinate and Fourier space. For various examples the convergence is analyzed. The results are applicable to various physics problems that use an expansion in a large number of oscillator states.