# Nonlinear Approximations in the Theory of Electron Detachment

Chapter

## Abstract

In Section 9.1 we considered a Hamiltonian with a separable potential multiplied by a coefficient linearly dependent on time, and showed that the solution of the corresponding time-dependent equation could be obtained with the help of contour integration. An equation where the coefficient of the separable potential is a linear function of time can be solved in a similar way. In the particular case when the coefficient of the separable potential is inversely proportional to time, the equation
has a solution of the form
which appears to be even simpler than the form of equation (9.1.4). Unfortunately, the perturbation term in (10.1.2) is singular at t = 0, which makes it difficult to apply this equation to real systems. The general behavior of the terms for a simplified problem which has only three discrete energy levels is illustrated in Fig. 10.1. In certain cases when the terms which go to infinity may be ignored, this model can still be used. In the particular case of the time-dependent equation
there is a system of exponentially diverging terms which interact with a single horizontal term. Making the substitution exp(at) = s, we transform equation (10.1.3) to the form (10.1.1); hence, the exact solution of equation (10.1.3) can be obtained making use of equation (10.1.2). A two-state system of this type was considered by Nikitin and also by Demkov (for references see (26) ), and it was applied to problems of non-resonant charge exchange and to the calculation of the fine-structure transitions in alkali metals caused by impact.

$$\rm \Big[H_0+\mid \phi >\gamma\ t^{-1}<\phi \mid \Big]\ \ \mid \psi >\ =\ i\ {\partial\over \partial t}\mid \psi >$$

(10.1.1)

$$\rm \mid \psi >\ =\ N{\int\limits_C}\ G(E)\ \mid \phi >\ exp\ \Big\{i\ {\int\limits^E}\ <\phi\ \mid G({E^\prime})\mid \phi >\ d{E^\prime}\ -\ Et\Big\}\ dE,$$

(10.1.2)

$$\rm \Big(H_0\ e^{at}\ +\ \mid \phi >\ b\ <\ \phi \mid\ \Big)\ \mid\ \psi >\ =\ i\ {\partial\over \partial t}\ \mid\ \psi\ >,$$

(10.1.3)

## Keywords

Total Cross Section Momentum Distribution Quadratic Approximation Nonlinear Approximation Hankel Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Plenum Press, New York 1988