On Siegert pseudostates: formal concepts and s-wave illustrations in one- and two-channel central potentials

in oneand two-channel central potentials Abstract Quantum scattering theory aims to model the interactions of particles in the quantum regime as the particles scatter from one another. The theoretical understanding of these events often necessitates solutions to the time-independent Schrödinger equation. For computational purposes, it becomes necessary to limit consideration of a scattering event to a finite volume of space. It is then crucial that the solutions to the time-independent Schrödinger equation obey a valid boundary condition at the boundary of the volume. In this study, solutions are forced to obey the Siegert boundary condition. The resulting wave functions are called Siegert pseudostates. Siegert pseudostates are a basis capable of representing physical scattering scenarios, including scenarios in which outgoing particle flux must escape the finite interaction volume. The application of the Siegert boundary condition eliminates the Hermiticity of the Hamiltonian, allowing for a complex eigenvalue, which results in an entirely discrete complex energy spectrum. Shape and Feshbach resonances are discussed and identified by the characteristically small imaginary part of their energy. A wave packet, serving as a model for a physical particle, is represented as a superposition of Siegert pseudostates, and allowed to evolve in time. Numerical investigations throughout the study validate the formalism.