Applying numerical continuation to the parameter dependence of solutions of the Schrödinger equation

Abstract

In molecular reactions at the microscopic level the appearance of resonances has an important influence on the reactivity. It is important to predict when a bound state transitions into a resonance and how these transitions depend on various system parameters such as internuclear distances. The dynamics of such systems are described by the time-independent Schrödinger equation and the resonances are modeled by poles of the S-matrix.Using numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, we are able to develop efficient and robust methods to study the transitions of bound states into resonances. By applying Keller's Pseudo-Arclength continuation, we can minimize the numerical complexity of our algorithm. As continuation methods generally assume smooth and well-behaving functions and the S-matrix is neither, special care has been taken to ensure accurate results.We have successfully applied our approach in a number of model problems involving the radial Schrödinger equation.

Numerical continuation of bound and resonant states of the two-channel Schrödinger equation

“…In [19] it is shown that this procedure does not introduce false solutions, that is, the zeros of F are precisely the poles of S, and that it eliminates any singularities at k = 0. This is an extension of a similar result for the one-dimensional, singlechannel systems case [17,21].…”

“…In [1] it was found, for one-dimensional quantum systems, that it is possible to apply the continuation to a regularized function derived from the S-matrix because that function does satisfy the necessary smoothness conditions. For several realistic potentials the resonances were tracked as parameters in the system were varied.…”

Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations

“…For single-channel radial problems, as in section 2.1, it was proposed in [1] to use a regularized function rather than the S-matrix itself to converge and to track down bound states and resonances. This function was defined as, per l-channel,…”

“…Their properties are summarized as follows. (1) The bound states are situated on the positive imaginary axis and correspond to real, negative (rel. to potential asymptote) energies.…”

“…In [1] we developed a framework for applying numerical continuation techniques in the context of bound states and resonances in spherically symmetric short-range potentials. We have shown that numerical continuation methods, originally developed in the study of dynamical systems, can be applied successfully to track bound and resonant states efficiently in terms of a varying system parameter.…”