We consider the second order linear differential equation

    \[y''=\left[{\Lambda^2\over t^\alpha}+g(t)\right]y,\]

where \Lambda is a large complex parameter and g is a continuous function. In previous works we have considered the case \alpha\in(-\infty,2] and designed a convergent and asymptotic method for the solution of the corresponding initial value problem with data at t=0. In this paper we complete the research initiated in those works and analyze the remaining case \alpha\in(2,\infty). We use here the same fixed point technique; the main difference is that for \alpha\in(2,\infty) the convergence of the method requires that the initial datum is given at a point different from the origin; for convenience we choose the point at the infinity. We obtain a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large \Lambda (not of Poincar\’e-type) of the solution of the problem. The generalization to non-linear problems is straightforward. An application to a quantum mechanical problem is given as an illustration.





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Ferreira, Chelo, López, José L., Sinusia, Ester Pérez