NewL2approach to quantum scattering: Theory

Abstract

By exploiting the soluble infinite tridiagonal (Jacobi)-matrix problem generated by evaluating a zeroth-order scattering Hamiltonian Ho in a certain L basis set, we obtain phase shifts, wave functions, etc. , which are exact for a full Hamiltonian H in which only the potential V is approximated, Only bound-bound (L ) matrix elements of the Hamiltonian and finite matrix manipulations are needed. The method is worked out hex e for s-wave scattering using Laguerre basis functions. Kato improvement of the results and necessary generalizations to many channels are treated.

Algebraic model for quantum scattering: Reformulation, analysis, and numerical strategies

“…Such a relationship between the wave function in coordinate space and its oscillator representation was already obtained long ago [4], [1] for the regular and irregular asymptotic solutions. Later, this correspondence has been used as a heuristic principle for solving e.g.…”

“…Ψ (+) is commonly called the "regular" solution, and behaves properly for all r. Ψ (−) is the "irregular" solution, and has an irregular (infinite) behaviour near the origin r = 0. To provide a regular character at the origin for both Ψ (+) and Ψ (−) , we redefine their equations in the following way, as was suggested earlier by Heller and Yamani [1]:…”

“…The kinetic energy contribution has a very simple representation in terms of the oscillator basis: (1) it is proportional to the inverse square of the oscillator radius b, and (2) the kinetic energy matrix has a tri-diagonal form, i.e. has non-zero matrix elements only along the major diagonal and the first subdiagonals.…”

“…The analysis of the AM equations presented in this work will be shown to lead to (1) an optimal choice for the basis, given the potential, yielding well-converging and stable solutions of the AM system, or, if the optimal basis cannot be used, to (2) algorithmic procedures for solving the AM system in an acceptable and controlled approximation. These algorithmic procedures will depend on the specific choice made for the basis (i.e.…”

Algebraic model for scattering in three-s-cluster systems. I. Theoretical background

“…The use of a basis of square integrable states reduces the Schrödinger equation to a matrix equation. This procedure is well-known for bound states, but is also applicable to continuum states when the appropriate boundary conditions are imposed on the expansion coefficients [1,2,[12][13][14]. Thus the Algebraic Model provides a unified approach to bound and continuous spectra based on familiar matrix techniques.…”