Thanks. This is what I tried to do and that lead to finding some bugs in the bessel functions. I know for E=0 it is not square integral but I was hoping to eventually find a way to express the Energy eigenvalues in terms of the bessel functions. (Probably can't be done, but anyway I know there are bessel transforms similar to the Fourier transform and Fourier series.)
Rich
------------Original Message------------
From: Barton Willis <willisb at unk.edu>
To: "Richard Hennessy" <rvh2007 at comcast.net>
Cc: "Maxima List" <maxima at math.utexas.edu>
Date: Sat, Jun-14-2008 6:09 PM
Subject: Re: [Maxima] QM
-----maxima-bounces at math.utexas.edu wrote: -----
>I noticed that in Maxima the contrib_ode package and the odelin function
>return for the x^4 potential always a complex valued combination of 2
>functions. I have been trying to find the corresponding real valued
>versions. This is a problem. The solutions have to be square integrable
>and it looks to me like f(z) and g(z) are but not so for f'(x) and g'(x)
>so how would you do this change of form? Is this a flaw in the proof?
Other than the zero function, no solution to -diff(y,x,2) + x^4 * y = 0 is
square integrable on (-inf, inf). If you need solutions to -diff(y,x,2) +
x^4 * y = 0
that are real on the real axis, it's not a problem:
(%i35) odelin(-'diff(y,x,2) + x^4 * y,y,x);
(%o35)
{bessel_j(-1/6,-(%i*x^3)/3)*sqrt(-x),bessel_y(-1/6,-(%i*x^3)/3)*sqrt(-x)}
A solution that is real-valued on the real axis is
(%i36) first(%) + conjugate(first(%));
(%o36)
bessel_j(-1/6,-(%i*x^3)/3)*sqrt(-x)+conjugate(bessel_j(-1/6,-(%i*x^3)/3))*conjugate(sqrt(-x))
Barton