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International Journal of Quantum Chemistry

Resonances and irreversibility for Schrödinger evolution

Pavel Kurasov, Nils Elander, B. S. Pavlov

Published Jan 1, 1993• 1 citation
Publisher

Abstract

Abstract The relations between the Schrödinger and wave evolutions are investigated. A generalization of the Lax‐Phillips scattering theory for the case of the Schrödinger equation is constructed. An analog of the wave decay operator is obtained. The general ideas are illustrated by an explicitly solvable model of multichannel scattering on the halfaxis. A Lax‐Phillips scattering theory is developed for the model operator obtained by a self‐adjoint perturbation of the background operator. © 1993 John Wiley & Sons, Int.

DOI
10.1002/qua.560460308

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Abstract The relations between the Schrödinger and wave evolutions are investigated. A generalization of the Lax‐Phillips scattering theory for the case of the Schrödinger equation is constructed. An analog of the wave decay operator is obtained. The general ideas are illustrated by an explicitly solvable model of multichannel scattering on the halfaxis. A Lax‐Phillips scattering theory is developed for the model operator obtained by a self‐adjoint perturbation of the background operator. © 1993 John Wiley & Sons, Int.
We discuss the quantum Lax–Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax–Phillips S-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup associated with resonances, the decay law is exactly exponential. We explain how this profound difference between the quantum Lax–Phillips theory and the description of unstable systems in the framework of the standard quantum theory emerges. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax–Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. In the special case of a time-independent potential problem lifted trivially to the quantum Lax–Phillips theory, we show that the Lax–Phillips S-matrix is unitarily related to the S-matrix of standard scattering theory by a unitary transformation parametrized by the spectral variable σ of the Lax–Phillips theory. Analytic continuation in σ has some of the properties of a method developed some time ago for application to dilation analytic potentials. We work out an illustrative example of the theory using a Lee–Friedrichs model, which is generalized to a rank one potential in the Lax–Phillips Hilbert space.
Basic notions and theWigner–Weisskopf theory Although the notions of bound states, scattering and quantum transitions are well defined in the quantum theory, the description of an unstable system, involved in the process of decay, has remained an outstanding issue for many years. The problem is fundamental, since it concerns the nature of irreversible processes, one of the most important issues in statistical mechanics and the theme that is central to this book. The theory of decay is intimately connected with scattering theory and necessarily contains mathematical ideas and methods. We shall try to explain these points carefully as we get to them. We treat elsewhere in the book the ideas of Boltzmann, Van Hove and Prigogine on irreversible phenomena. The tools that are developed there are basically approximate, although very useful. One can argue that the basic rigorous characteristic of an irreversible process is that, as represented in terms of the evolution of a state in the Hilbert space of the quantum theory, it must be a semigroup. This type of evolution, resulting in an operation Z ( t ) on a state ψ , should satisfy the property Z ( t 2 ) Z ( t 1 ) = Z ( t 1 + t 2 ). (17.1) The argument is as follows. If the system evolves in time t 1 and is stopped, then evolves further at time t 2 , since the process has no memory, the total evolution should be as if the system evolved from the initial state to a state at t 1 + t 2 independently of the fact that it was done in two stages (Piron, 1976). Since the process is irreversible, the operator Z ( t ) may have no inverse. Such an evolution is called a semigroup . As we shall see, it is not possible to obtain such an evolution law in the framework of the standard quantum theory (Horwitz et al. , 1971), but recently much work has been done, and methods have been developed, based on ideas of Sz.-Nagy and Foias (1976), such as the theory of Lax and Phillips (1967) and its extension to the quantum theory (Strauss et al. , 2000) in which semigroup evolution can be achieved.
It has been shown by Flesia and Piron that the scattering theory of Lax and Phillips, developed primarily for the treatment of hyperbolic systems (e.g., classical scattering of electromagnetic waves on a finite target), can be applied to the quantum theory. In order to construct the correspondence, the continuously infinite family of (isomorphic) Hilbert spaces associated with each value of the time (are taken to be a direct integral representation (with a choice of measure) of a larger Hilbert space which includes the variable t in its measure space. We show that this theory, for which the evolution contracted to the subspace orthogonal to the incoming and outgoing subspaces is a contractive semigroup, provides a natural description of the irreversible motion of an unstable quantum system
The scattering theory of Lax and Phillips, designed primarily for hyperbolic systems, such as electromagnetic or acoustic waves, is described. This theory provides a realization of the theorem of Foias and Nagy; there is a subspace of the Hilbert space in which the unitary evolution of the system, restricted to this subspace, is realized as a semigroup. The embedding of the quantum theory into this structure, carried out by Flesia and Piron, is reviewed. We show how the density matrix for an effectively pure state can evolve to an effectively mixed state (decoherence) in this framework. Necessary conditions are given for the realization of the relation between the spectrum of the generator of the semigroup and the singularities of the SSS-matrix (in energy representation). It is shown that these conditions may be met in the Liouville space formulation of quantum evolution, and in the Hilbert space of relativistic quantum theory.
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: