Expansion Approach to Scattering

Abstract

We describe an expansion method for calculating scattering phase shifts which avoids the difficulties of the variational methods of Hulthen 1 and Kohn. 2 Demkov and Shepelenko 3 have pointed out the algebraic source of these difficulties, and they have shown how the Kohn and Hulthen methods implicitly eliminate an equation to restore consistency. These methods have also been examined elsewhere. 4 ? 5 A third variational method, due to Schwinger, 6 appears to be more difficult to use, although Schwartz 7 has indicated a way to apply Schwinger's method for potential scattering, and Lippmann 8 has recently converted the Schwinger principle into a form similar to those of Hulthen and Kohn. Other approaches are those of Schlessinger and Schwartz, 9 and of Spruch, Rosenberg, Hahn, and O'Malley. 10 The former proceeds by analytic continuation from boundstate regions, while the latter simplifies calculations for compound targets.Ours is an approach which seeks to use to advantage the singularity properties that adversely affect the Kohn method. As it stands, it is applicable to compound targets, and it appears extendable to multichannel processes. Consider a single-channel process with Hamiltonian H and asymptotic solutions ip t (E) and $ 2 (E) at energy E. The stationary-state wave function^ is then written \£ = a 1 $ l + a 2 il) 2 + <i >, thereby defining $, and the Schrodinger equation yields aAH-E)il>AE)+a (H-E)ipJE) + (H-E)<!> = 0. (I) 1 1 z IWe assume H to be such that at large distances the incident particle sees a short-range potential, so that (H-E)r\> ly (F-E)^2, (H-E)$, and $ are all quadratically integrable. The solution of Eq. (1) for $ then should be expected to be smoothly approximable with boundstate functions. Introduce a set of bound-state functions X(, i-1, • • * 9 n, to be used for the approximation of $. The role of these functions is most clearly seen if they are transformed to a basis on which H is diagonal within the subspace spanned by the x;-That is, we construct and solve the finite matrix equation (H-A §) c = 0, where Hy = (xflXj) and S t j = (xtXj)-The solution 5^, cor-responding to eigenvalue A ", defines a function (P^TJV 0 viiXp* Now, suppose that the functions cp ^ are capable of giving a good representation of $ at an energy E. Then Eq. (1) should be nearly satisfied. The condition we shall impose here is that the left side of Eq. (1) have no component in the subspace spanned by the cp . Because of the bounded nature of all terms of Eq.(1), this condition should produce convergence to a well-defined solution as the cp " approach a complete set of quadratically integrabe functions.The essence of our method is to recognize that the above-described condition is easy to impose if E is chosen to be an eigenvalue A" of the finite probelm which we used to define the cp ". For E = X the coefficient of cp " on the left of Eq. (1) is fl l^'%'* 1 V >+8 2 < % l % , W > '

Integral relations and the adiabatic expansion method for1+2reactions above the breakup threshold: Helium trimers with soft-core potentials

“…When E > 0 (such as E (3) in Fig. 1), the breakup channel is also open, and it is described by the remaining infinitely many adiabatic potentials.…”

“…1 for E = E (3) ). Of course, still a finite number of them correspond to elastic, inelastic, or transfer processes, and the infinitely many remaining ones describe the breakup channel.…”

“…y → κ, and y n → ρ have to be made [note that the relations j K+ 3 2 (z) = π 2z J K+2 (z) and η K+ 3 2 (z) = π 2z Y K+2 (z) permit us to write Eqs. (12) and (13) in terms of the Bessel functions J K+2 and Y K+2 , which is how the asymptotic form of the breakup channels is usually presented in the literature].…”

“…As shown in [1,2], this can indeed be done by means of two integral relations that are based on the Kohn variational principle. These two integral relations are a generalization to more than two particles of the integral relation given in [3,4], and they permit us to obtain the K (or S) matrix of the reaction by using only the internal part of the wave function. Therefore, all the physical information concerning a given 1 + N reaction can be obtained without an accurate knowledge of the asymptotic part of the wave function.…”

A simple method for evaluating low-energy electron-molecule scattering cross sections using discrete basis functions

“…In general, it is necessary to develop a formula for the phaseshift which is independent of the normalization of the wavefunction. For example, the Harris formula (Harris 1967) can be viewed in this manner. These methods are unfortunately not rigorously applicable to scattering from a non-spherical potential.…”