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The Journal of Chemical Physics

Variational approach to the quantum theory of scattering. II. The Kohn method and Kato eigenvalue problem

H. J. Kolker

Published Mar 15, 1973• 6 citations
Publisher

Abstract

Certain similarities between the equations arising in scattering theory and the variational approach to time dependent perturbation theory are used to show that the Kohn variational technique for solving scattering problems is equivalent to a variational method for calculating a special set of eigenfunctions, in terms of which the solution to the scattering problem is expanded. Contrary to the results of previous studies, the error term in the scattering problem is bound from below, and the Kohn method is equivalent to a minimum principle. Levinson's theorem is rederived, and extended to higher energies. On the basis of the theory presented in this study, a new method for calculating upper and lower bounds on the energies of resonance and Ramsauer-Townsend states is suggested.

DOI
10.1063/1.1679504

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Certain similarities between the equations arising in scattering theory and the variational approach to time dependent perturbation theory are used to show that the Kohn variational technique for solving scattering problems is equivalent to a variational method for calculating a special set of eigenfunctions, in terms of which the solution to the scattering problem is expanded. Contrary to the results of previous studies, the error term in the scattering problem is bound from below, and the Kohn method is equivalent to a minimum principle. Levinson's theorem is rederived, and extended to higher energies. On the basis of the theory presented in this study, a new method for calculating upper and lower bounds on the energies of resonance and Ramsauer-Townsend states is suggested.
It is pointed out that the notation of the theory of functionals and the theorem in it analogous to Taylor's theorem facilitates a compact discussion of perturbation and variation methods in quantum mechanics, both for bound state and scattering problems. With this aid it is shown (i) how stationary expressions required for variation methods may be sought and found, (ii) that the perturbation and variation methods are virtually identical, (iii) that second order perturbations and expressions stationary to the second order may be found similarly, though these are complicated. The form found for the second order perturbation of bound states differs from that commonly used in that it does not explicitly involve the eigenfunctions of states other than that perturbed.
Throughout our discussion of the general principles of quantum mechanics we have emphasized the importance of the eigenvalues and eigenfunctions of the operators representing physical quantities. However, we have seen that solving the basic eigenvalue equations determining these is often not straightforward. In the early chapters discussing the energy eigenvalue equation (or time-independent Schro¨dinger equation) for example, we found that this could often not be solved exactly and that, even when a solution was possible, it frequently required considerable mathematical analysis. Because of this, methods have been developed to obtain approximate solutions to eigenvalue equations. One of the most important of such techniques is known as time-independent perturbation theory and will be the first method to be discussed in the present chapter. Another, known as the variational principle, will be discussed later. We shall confine our discussion to the timeindependent Schro¨dinger equation, but generalization to other eigenvalue equations is quite straightforward. We shall mainly use the wave function representation, but we shall restate some of the important results in Dirac notation so that they can be readily transformed to a matrix representation if desired.
The scattering length AAA characterizes the zero-energy scattering of one system by another. It was shown some time ago that a variational upper bound on AAA could be obtained using methods, of the Rayleigh-Ritz type, which are commonly employed to obtain upper bounds on energy eigenvalues. Here we formulate a method for obtaining a variational lower bound on AAA. Once again the essential idea is to express the scattering length as a variational estimate plus an error term and then to reduce the problem of bounding the error term to one involving bounds on energy eigenvalues. In particular, the variational lower bound on AAA is rigorously established provided a certain modified Hamiltonian can be shown to have no discrete states lying below the level of the continuum threshold. It is unfortunately true that necessary conditions for the existence of bound states are not available for multiparticle systems in general. However, in the case of positron-atom scattering the adiabatic approximation can be introduced as an (essentially) solvable comparison problem to rigorously establish the nonexistence of bound states of the modified Hamiltonian. It has recently been shown how the validity of the variational upper bound on AAA can be maintained when the target ground-state wave function is imprecisely known. Similar methods can be used to maintain the variational lower bound on AAA. Since the bound is variational, the error in the calculated scattering length will be of second order in the error in the wave function. The use of the adiabatic approximation in the present context places no limitation in principle on the accuracy achievable.
The treatment of Schrödinger's perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. The solution of the Schrödinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an alternative insight in the mathematical structure of perturbation theory, particularly with respect to the ``bracketing theorem'' and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately.
In chapters 13 and 14 we focused on perturbation theories for time-independent and time-dependent perturbations. For time-independent perturbations we obtained first-order and second-order corrections to energy eigenvalues and eigenfunctions. For time-dependent perturbations we obtained the probability for a system to make a transition from an ith initial state to fth final state. We have also discussed how to study the effect of adiabatic and sudden perturbations. In the present chapter we consider the variational method. The variational method can be used to construct an approximate ground state eigenfunction. In some cases the method can be used for finding approximately the energy of first few discrete states of a bound system. The perturbation theory uses a set of unperturbed states and the perturbation Hamiltonian H(1) to construct the eigenfunctions and eigenvalues through highly formalized approximations. In contrast, the variational method is based on a guess for the state-a trial state which is used to determine the corresponding energy. Any well behaved trial state introduced will yield an appropriate energy, although some trial states will be better than other, but whatever the guess, we can always make systematic improvements. We can construct a reasonably accurate eigenfunction often by a physical insight and ingenuity.