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let be the differential equation $ -y''(x)+x^{4}y(x)-E_{n}y(x)=0 $ with the boundary conditions $ y(0)=0=y(\infty) $

how could i use the shooting method or other numerical method to solve this equation ? , my only idea is to set $ R=10000 $ for example and solve $ y(0)=0=y(R) $

of course i also could made the substitution $ u= \frac{x}{x-1} $ so the new boundary conditions could become $ y(0)=0=y(1) $ but know the differential equation would be singular at the point $ u=1 $

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2 Answers

up vote 2 down vote accepted

You're facing a singular Sturm-Liouville eigenvalue problem. One way you may be able to use the shooting method is to get an asymptotic expansion of your solution at the $x=\infty$ (Frobenius or WKB methods should work). Using the asymptotics to evaluate $y(x)$ and $y'(x)$ at some finite $x$ and then shoot from there to $x=0$ to find the eigenvalue.

On the other hand, there exists a fairly well developed numerical library, based also on a sound mathematical analysis of the Sturm-Liouville problem, that is already available. It should be able to handle your problem directly. Look up the SLEIGN2 package and its documentation.

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See the recent preprints of Andre Voros on the arXiv. He proposed an iterative algorithm, and Artur Avila recently proved convergence. The literature on this problem is enormous; it usually goes under the title "anharmonic oscillator" or "quartic oscillator". The rate of convergence of Voros algorithm is geometric.

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