In the present paper we shall be concerned with the stationary theory of scattering associated with the Schrodinger operator with a long-range potential. In quantum theory of scattering, many authors have investigated the existence and completeness of wave operators W = s-lim ee~y where J-»±oo HI and H2 are self-adjoint operators acting on a Hilbert space M. Among them, Kato and Kuroda gave an abstract time-independent approach to scattering theory. They derived a stationary form of the wave operator in the following way:In the present paper we shall be concerned with the stationary theory of scattering associated with the Schrodinger operator with a long-range potential. In quantum theory of scattering, many authors have investigated the existence and completeness of wave operators W = s-lim ee~y where J-»±oo HI and H2 are self-adjoint operators acting on a Hilbert space M. Among them, Kato and Kuroda gave an abstract time-independent approach to scattering theory. They derived a stationary form of the wave operator in the following way:

Publications of the Research Institute for Mathematical SciencesDec 31, 197711 citations

Stationary Theory of Scattering

T. Kato

Springer eBooksJan 1, 19771 citation

Potential scatterings in $L^2$ space: (1) non-orthogonality of stationary statesPotential scatterings in $L^2$ space: (1) non-orthogonality of stationary states

Kenzo Ishikawa

Orthogonality of eigenstates of different energies and its implications in potential scattering are unlabeled. Scalar products of scattering states of different energies are found to have finite non-orthogonal terms in potentials of finite widths. Their superpositions have time-dependent norms, and are not suitable for isolate states. In these systems, a perturbative method and a variational method are viable methods for finding a rigorous transition probability that describes phenomena completely. In various exceptional potentials, an orthogonality is satisfied.Orthogonality of eigenstates of different energies and its implications in potential scattering are unlabeled. Scalar products of scattering states of different energies are found to have finite non-orthogonal terms in potentials of finite widths. Their superpositions have time-dependent norms, and are not suitable for isolate states. In these systems, a perturbative method and a variational method are viable methods for finding a rigorous transition probability that describes phenomena completely. In various exceptional potentials, an orthogonality is satisfied.

arXiv:2305.16939May 26, 20230 citations

Universal Preparability of States and Asymptotic Completeness

Rolf Gohm, Florian Haag, Burkhard Kümmerer

We introduce a notion of universal preparability for a state of a system, more precisely: for a normal state on a von Neumann algebra. It describes a situation where from an arbitrary initial state it is possible to prepare a target state with arbitrary precision by a repeated interaction with a sequence of copies of another system. For ${\mathcal{B}(\mathcal{H})}$ we give criteria sufficient to ensure that all normal states are universally preparable which can be verified for a class of non-commutative birth and death processes realized, in particular, by the interaction of a micromaser with a stream of atoms. As a tool, the theory of tight sequences of states and of stationary states is further developed and we show that in the presence of stationary faithful normal states universal preparability of all normal states is equivalent to asymptotic completeness, a notion studied earlier in connection with the scattering theory of non-commutative Markov processes.We introduce a notion of universal preparability for a state of a system, more precisely: for a normal state on a von Neumann algebra. It describes a situation where from an arbitrary initial state it is possible to prepare a target state with arbitrary precision by a repeated interaction with a sequence of copies of another system. For $${\mathcal{B}(\mathcal{H})}$$ we give criteria sufficient to ensure that all normal states are universally preparable which can be verified for a class of non-commutative birth and death processes realized, in particular, by the interaction of a micromaser with a stream of atoms. As a tool, the theory of tight sequences of states and of stationary states is further developed and we show that in the presence of stationary faithful normal states universal preparability of all normal states is equivalent to asymptotic completeness, a notion studied earlier in connection with the scattering theory of non-commutative Markov processes.

arXiv:1605.00315Communications in Mathematical PhysicsMar 6, 20172 citations

Potential scatterings in

L^2

space: (1) non-orthogonality of stationary states

{\mathcal{B}(\mathcal{H})}

we give criteria sufficient to ensure that all normal states are universally preparable which can be verified for a class of non-commutative birth and death processes realized, in particular, by the interaction of a micromaser with a stream of atoms. As a tool, the theory of tight sequences of states and of stationary states is further developed and we show that in the presence of stationary faithful normal states universal preparability of all normal states is equivalent to asymptotic completeness, a notion studied earlier in connection with the scattering theory of non-commutative Markov processes.