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Physical Review Letters

New Variational Principle for Decaying States

C. F. Hart, Marvin D. Girardeau

Published Nov 7, 1983• 5 citations
Publisher

Abstract

The Hamiltonian of a system containing decaying states is subjected to a nonunitary similarity transformation which introduces the wave functions of these decaying states as well as those of the "antidecaying" dual states explicitly into the representation. The dependence of the Green's-function pole on these wave functions is used to formulate a variational principle for their determination, yielding a non-Hermitian, nonlinear wave equation including self-energy contributions. Explicit solutions are given in a simple example.

DOI
10.1103/physrevlett.51.1725

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The Hamiltonian of a system containing decaying states is subjected to a nonunitary similarity transformation which introduces the wave functions of these decaying states as well as those of the "antidecaying" dual states explicitly into the representation. The dependence of the Green's-function pole on these wave functions is used to formulate a variational principle for their determination, yielding a non-Hermitian, nonlinear wave equation including self-energy contributions. Explicit solutions are given in a simple example.
We used nonequilibrium many-body Green techniques for the time varying resonant phenomena in few-body systems. By relating the transition energy shift and width to the self-energy and by applying the functional variational techniques to the transition energy shift and width, the corresponding initial state resonant wave functions are derived. Then we obtained time-dependent decaying state wave functions, evolving from these initial states by solving the time-dependent Schr\"odinger equation for the decaying system numerically. Finally, we observed the plausible time decaying behaviors of our model as predicted by real few-body systems.
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 .
A variational principle is formulated which is invariant with respect to normalization of wave functions and which can be used in the case of nonorthogonal wave functions of the initial and final states. The problera of uniqueness of perturbation theory in this case is considered. The possibility of employing the variational principle for solution of problems concerning decay of the quasi-stationary state and charge exchange is discussed. The main equations for charge exchange at large impact parameters are derived from the variational principle. (auth)