In recent work, the existence of stationary bounds on bound-state matrix elements for rather broad classes of quantum-mechanical operators was used to obtain a stationary upper bound on the scattering length for the collision of positrons with atoms whose target wave functions are not precisely known. This was done by obtaining calculable stationary bounds on the unknown matrix elements which appear in the formal expression for the bound on the scattering length. This method is extended in the present paper to electron-atom scattering for which it is necessary to obtain-in addition to bounds of the previous type-stationary bounds on the various exchange integrals. These latter cannot be obtained by means of techniques previously described. New techniques are presented for deriving such bounds, which are then used to obtain rigorous stationary upper bounds on the scattering length for the collisions of electrons with helium and with atoms heavier than helium.In recent work, the existence of stationary bounds on bound-state matrix elements for rather broad classes of quantum-mechanical operators was used to obtain a stationary upper bound on the scattering length for the collision of positrons with atoms whose target wave functions are not precisely known. This was done by obtaining calculable stationary bounds on the unknown matrix elements which appear in the formal expression for the bound on the scattering length. This method is extended in the present paper to electron-atom scattering for which it is necessary to obtain-in addition to bounds of the previous type-stationary bounds on the various exchange integrals. These latter cannot be obtained by means of techniques previously described. New techniques are presented for deriving such bounds, which are then used to obtain rigorous stationary upper bounds on the scattering length for the collisions of electrons with helium and with atoms heavier than helium.

Physical Review AJan 1, 197511 citations

Non-Stationary Quantum Systems Associated with Stationary Scattering in One Dimension

Ali Mostafazadeh, Alireza Shariati Joni

Stationary scattering admits a dynamical formulation in which the scattering data are extracted from the evolution operator for an effective non-stationary and non-unitary quantum system. This provides a mathematical method of solving scattering problems. We consider the possibility of realizing such a quantum system and relating the scattering information to outcomes of quantum measurements performed on it. To this end, we examine stationary scattering problems in one dimension where the scattering data consist of reflection and transmission amplitudes and the corresponding effective quantum system is a non-unitary two-level system. The states of this system identify points in a Bloch sphere $S^2$ and their evolution yields trajectories in $S^2$ . Quantum measurements turn out to be capable of providing information about the reflection amplitudes. We identify configurations displaying reflectionlessness with situations where one or both of the poles of $S^2$ undergo cyclic evolutions. Spectral singularities and coherent perfect absorption correspond to dynamical spin flips which map one of the poles of $S^2$ to the other.Stationary scattering admits a dynamical formulation in which the scattering data are extracted from the evolution operator for an effective non-stationary and non-unitary quantum system. This provides a mathematical method of solving scattering problems. We consider the possibility of realizing such a quantum system and relating the scattering information to outcomes of quantum measurements performed on it. To this end, we examine stationary scattering problems in one dimension where the scattering data consist of reflection and transmission amplitudes and the corresponding effective quantum system is a non-unitary two-level system. The states of this system identify points in a Bloch sphere $S^2$ and their evolution yields trajectories in $S^2$. Quantum measurements turn out to be capable of providing information about the reflection amplitudes. We identify configurations displaying reflectionlessness with situations where one or both of the poles of $S^2$ undergo cyclic evolutions. Spectral singularities and coherent perfect absorption correspond to dynamical spin flips which map one of the poles of $S^2$ to the other.

quant-pharXiv:2610.15655Oct 13, 20260 citations

S^2

and their evolution yields trajectories in

S^2

. Quantum measurements turn out to be capable of providing information about the reflection amplitudes. We identify configurations displaying reflectionlessness with situations where one or both of the poles of

S^2

undergo cyclic evolutions. Spectral singularities and coherent perfect absorption correspond to dynamical spin flips which map one of the poles of

S^2

to the other.