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After spending a couple of weeks, I was able to find the solution to a certain differential equation, given below (Well they are the eigenfunctions to be exact):

$$y_n(x) = \frac{e^{ax^2}}{\sqrt{x}}\cdot \prod_{i=1}^n \left( \frac{1}{\sqrt{x}}-t_i \right) $$

where $a\in \mathbb{R}^-$ and the $t_i$'s are constants.

I wish to ask the question whether anyone knows a way to find $$\int_{\alpha}^{\infty}\left|y_n(x)\right|^2$$

Where $\alpha>0$

As it would be nice if I could find a normalization constant for this function. Is there a way that this can be done with contour integration or Feynman's parametrization trick?

share|cite|improve this question
Where is this problem coming from ? – Claude Leibovici Jan 16 '14 at 7:03
I've modified it. I remembered normalization does not have to be strictly in the range $[0,\infty)$ – Millardo Peacecraft Apr 4 '14 at 21:26

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