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.
One recipe for mathematically formulating a relativistic quantum mechanical scattering theory utilizes a two-Hilbert space approach, denoted by
H and
H0, upon each of which a unitary representation of the Poincaré Lie group is given. Physically speaking,
H models a complicated interacting system of particles one wishes to understand, and
H0 an associated simpler (i.e., free/noninteracting) structure one uses to construct 'asymptotic boundary conditions" on so-called scattering states in
H. Simply put,
H0 is an attempted idealization of
H one hopes to realize in the large time limits
t→±∞. The above considerations lead to the study of the existence of strong limits of operators of the form
eiHtJe−iH0t, where
H and
H0 are self-adjoint generators of the time translation subgroup of the unitary representations of the Poincaré group on
H and
H0, and
J is a contrived mapping from
H0 into
H that provides the internal structure of the scattering asymptotes. The existence of said limits in the context of Euclidean quantum theories (satisfying precepts known as the Osterwalder-Schrader axioms) depends on the choice of
J and leads to a marvelous connection between this formalism and a beautiful area of classical mathematical analysis known as the Stieltjes moment problem, which concerns the relationship between numerical sequences
{μn}n=0∞ and the existence/uniqueness of measures
α(x) on the half-line satisfying \begin{equation*} \mu_{n}=\int_{0}^{\infty}x^{n}d\alpha(x). \end{equation*}
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.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
A general framework is set up for the description of quantum-mechanical scattering processes for the case that both the free hamiltonian H 0 (t)=H 0 +H 1 (t) and the full hamiltonian H(t)=H 0 (t)+V are time-dependent. In particular the existence of the various evolution operators is considered and the wave operators Omega +or- (s), which now depend on the initial time s, are defined. The consequences of a periodic time dependence are studied and for H 1 (t) of the type H 1 (t)=H 1 cos( omega t+ delta ) it is shown that for mod omega mod to infinity the frequency-dependent wave operators converge to the wave operators which pertain to the case that H 1 (t) is absent.
The mathematical theory of the scattering operator is developed for the general scattering systems involving an arbitrary number of channels. It includes as a special case the theory for 'simple scattering systems' given in an earlier paper. The scattering system is defined as a quantum mechanical system which satisfies certain asymptotic and completeness conditions given in Section 4. The existence of the S-operator as well as its unitary property is then a rigorous mathematical consequence of this property. A crucial step in these deductions is the orthogonality theorem for the left projections of the wave operators which is proved in Section 5. In the last Section 7, we discuss the various ways of introducing the 'in' and 'out'-operators and their relation to the S-operator.