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Springer eBooks

Irreversible Quantum Evolution for Unstable Systems in Lax-Phillips Scattering Theory

Lawrence Paul Horwitz, Eli Eisenberg, Y. Strauss

Published Jan 1, 1997• 0 citations
Publisher
DOI
10.1007/978-94-011-5886-2_23

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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

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Representation of quantum mechanical resonances in the Lax–Phillips Hilbert space

Y. Strauss, Lawrence Paul Horwitz, Eli Eisenberg

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.

arXiv:hep-th/9709036•Journal of Mathematical Physics•Dec 1, 2000•29 citations
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
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.