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Journal of Mathematical Physics

Resonances, scattering theory, and rigged Hilbert spaces

Guido Parravicini, Vittorio Gorini, E. C. G. Sudarshan

Published Aug 1, 1980• 95 citations

Abstract

The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,’’in,’’ and ’’out’’ eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ’’out’’ eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ’’complete’’ sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ’’out’’ eigenvectors. The free, ’’in’’ and ’’out’’ eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential.

DOI: 10.1063/1.524655

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The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,’’in,’’ and ’’out’’ eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ’’out’’ eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ’’complete’’ sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ’’out’’ eigenvectors. The free, ’’in’’ and ’’out’’ eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential.

Within the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self-adjoint Hamiltonian with complex eigenvalue and a Breit–Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.

After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S-matrix poles in the lower half of the second sheet and exponentially growing vectors from the S-matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.

The Hermitian properties of the Hamiltonian are related not only to the operator itself but also to the functions on which it acts. Hermitian Hamiltonians operate on functions in the L 2 Hilbert space which correspond to boundary conditions which vanish at infinity. In this chapter, in order to move into the non-Hermitian domain, we will impose on the solutions to the time-independent Schrödinger equation (TISE) different boundary conditions which lead to solutions which can be associated with different types of the complex poles of the scattering matrix. These solutions will contain information which was not available within the scope of functions in L 2 . By imposing outgoing boundary conditions on the eigenfunctions of the timeindependent Hamiltonian complex eigenvalues, E res = ε − (i/2)Г, are obtained. These complex energies are associated with decaying resonance states which were discussed in the previous two chapters. The bound states (if they exist) appear as real eigenvalues since they result from exactly such outgoing boundary conditions which appear under the threshold energy. When incoming boundary conditions are imposed two kind of solution are obtained. One type of solution is the complex conjugates of the decay resonance solutions mentioned above. In scattering theory text books (see Taylor for example) the physical resonance solutions are associated with the poles of the scattering matrix which are embedded in the lower half of the complex energy plane. However, in nuclear physics the complex poles embedded in the upper half of the complex energy plane, so-called virtual states , are denoted as capture resonances .

The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states

ϕ^+

and the out-states (observables)

ψ^-

of the S-matrix elements

(ψ^-,ϕ^+)=(ψ^{out},S ϕ^{in})

any elements of the HS $\H$,

\{ψ^-\}=\{ϕ^+\}(=\H)

, the RHS theory chooses the boundary conditions~:

ϕ^+\inΦ_-\subset\H\subsetΦ_-^\times

,

ψ^-\inΦ_+\subset \H\subset Φ_+^\times

, where

Φ_-

(

Φ_+

) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states

ϕ^+

and observables

ψ^-

provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets

|E^{\pm}\inΦ_{\mp}^{\times}

(solutions of the Lippmann-Schwinger equation with

\pm iε

respectively) and the Gamow vectors

|E_R-iΓ/2^\pm\inΦ_{\mp}^\times

. The Gamow vectors

|E_R-iΓ/2^-

have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.

Traditional Quantum Scattering theory is built upon a Hilbert space formalism. The nature of the Hilbert space has led to a number of ideas about the nature of scattering systems, chief among them being the understanding that the decay of unstable quan- tum systems cannot be purely exponential and that there can only be an approximate relationship between the width Γ of a scattering resonance and the lifetime τ of the resonant state whose formation gives rise to it. That such physical conclusions are de- rived from purely mathematical properties of the underlying topological vector space inform the notion that, in choosing a particular space, one is choosing the mathemat- ical representation of the physical universe for the system. With this notion in mind, purely physical characteristics of quantum systems are used to answer the question: “What is the proper mathematical space for quantum mechanical calculations?” In the particular case of quantum scattering, it is shown that this approach casts the choice of the Hilbert space into doubt and motivates instead the use of rigged Hilbert spaces employing Hardy spaces; in these Hardy rigged Hilbert spaces, exponential decay is permissible and the time evolution of decaying states is governed by semigroups. The original theories concerning deviation from exponential decay in the Hilbert space are revisited to reveal how their conclusions flow from a mathematical property whose physical interpretation is problematic. Finally, an application of a Hardy rigged Hilbert space formalism is presented: Time Asymmetric Quantum Mechanics, as developed by A. Bohm, M. Gadella, and S. Wickramasekara.