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Applying numerical continuation to the parameter dependence of solutions of the Schrödinger equation. (English) Zbl 1190.65190

Summary: 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 H. B. Keller’s pseudo-arclength continuation [Application of bifurcation theory, Proc. adv. Semin., Madison/Wis. 1976, 359–384 (1977; Zbl 0581.65043)], 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.

##### MSC:
 65P30 Bifurcation problems (numerical analysis) 65H20 Global numerical methods for nonlinear algebraic equations, including homotopy approaches 65Y20 Complexity and performance of numerical algorithms 37M20 Computational methods for bifurcation problems 81U20 $S$-matrix theory, etc. (quantum theory)
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