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Physical Review A

Completeness relation in the R -matrix theory of scattering

Kazuo Takayanagi

Published Jun 23, 2008• 5 citations
Publisher

Abstract

We present the biorthogonal complement of the standing wave state and its relation to scattering operators. This establishes a completeness relation in terms of the standing wave states, and therefore completes the $\mathit{R}$-matrix (reactance matrix) theory of scattering. We clarify with an example that the $\mathit{R}$-matrix theory, in its formulation presented here, can claim its due place in addition to the standard $\mathit{T}$-matrix theory.

DOI
10.1103/physreva.77.062714

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We present the biorthogonal complement of the standing wave state and its relation to scattering operators. This establishes a completeness relation in terms of the standing wave states, and therefore completes the R\mathit{R}R-matrix (reactance matrix) theory of scattering. We clarify with an example that the R\mathit{R}R-matrix theory, in its formulation presented here, can claim its due place in addition to the standard T\mathit{T}T-matrix theory.
The most important physical aspects of quantum mechanics are interactions in bound systems and predictions relevant to particle scatterings. A special accent is placed upon the foundation of quantum scattering theory from the first principles of physics, without resorting to any free parameters. Such a complete theory, which is autonomously and consistently devised on a rigorous mathematical basis, represents a genuine modern candidate for a powerful predictor of real events in experiments. For modern scattering theory, the following two principal questions emerge: the existence and completeness of the Moller wave operators. They directly imply the unitarity of the S-matrix and the correctness of the boundary conditions. Such a general and versatile methodological concept largely surpasses the frames of scattering theory.
In this paper we give a complete theory of the scattering operator on a rigorous mathematical foundation for a certain class of quantum mechanical systems called, 'simple scattering systems'. These systems are characterized by three properties formulated in Section 3. The relation of these properties to the physical content of scattering is explained. We then proceed to define the wave operator and the scattering operator and prove their existence with mathematically rigorous methods. No assumptions are needed regarding the occurrence of bound states and the location of their energy with respect to the continuum region of the total energy operator. In Section 4 we demonstrate the existence of a certain integral representation of the scattering operator which has often been used as a starting point of the iteration approximation. Certain general properties of scattering systems are obtained in Section 5. This leads to necessary conditions for systems to be scattering systems as well as to a new definition of the scattering operator. The paper concludes with a derivation of the scattering cross-section from the scattering operator. The improvement of this derivation over the older ones sonsists in the special care employed for stating the assumptions which must go into such a derivation and the avoidence of improper functions or other artificial devices.
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We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a $2\ifmmode\times\else\texttimes\fi{}2$ matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. We show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices, each related to the time-evolution operator for an effective nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for $\ensuremath{\delta}$-function potentials in 2D and 3D and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.
A definition is formulated of the scattering matrix for a closed physical system with bound states which makes use throughout only of the assumed observable properties of the system. A direct product space, X, is defined in which the ingredient factor space comprises the steady states—vacuum, one-particle, and bound states—of the physical system. It is argued that the boundary conditions for a scattering experiment are suitably expressed in terms of vectors in X and that these stand in unitary correspondence, U, to the Heisenberg states. Indeed, one defines two operators U(±) to express outgoing and ingoing wave boundary conditions, and the scattering matrix is constructed from these in the usual way. A suitable Yang-Feldman formalism is then developed in which the operators in the remote past and future also describe the bound states of the system. A representation of the framework thus constructed in terms of field operators for individual fields results in the well-known formulas for S-matrix elements in terms of covariant amplitudes.
After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.