PseudoStates

Pseudostates are discrete energy levels obtained by solving a quantum system in a finite, square-integrable basis. They are not true eigenstates of the full Hamiltonian but representations of the continuum or unbound spectrum that have been discretized. When a system with unbound states is described within a finite basis or spatial domain, the continuum of scattering states is replaced by a countable set of pseudostates with corresponding energies. As the basis set is enlarged and better suited functions are used, the pseudostates approximate the true continuum, enabling approximate calculation of observables such as cross sections, transition rates, and response functions.

Construction methods typically involve choosing a complete L2 basis such as harmonic oscillators, B-splines, Gaussian bases, or grid-based approaches, and then diagonalizing the Hamiltonian in that basis. The resulting spectrum contains both bound-state levels and pseudostates above thresholds. Pseudostates are interpreted as representations of the density of unbound states rather than exact physical eigenstates. To extract physical quantities, smoothing procedures or continuum-discretization schemes are often employed.

Applications span multiple areas of quantum physics. In nuclear physics, pseudostates are used in reaction calculations and to treat breakup channels. In atomic and molecular physics, they assist in modeling photoionization and electron-scattering processes. In quantum chemistry, pseudostates can facilitate the inclusion of continuum effects in scattering and resonance problems.

Limitations include dependence on the chosen basis and finite-size effects, as well as convergence considerations. Pseudostates provide a practical, approximate representation of the continuum and require validation against benchmark results or converged calculations.