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I am doing research in the field of number theory and as part of this looking for correspondencies to other discilines and particularly physics. I am searching for examples in physics where the equation below finds its application: $$\frac{\partial^2\psi}{\partial x^2}=\psi(x)+ x \frac{\partial\psi}{\partial x} \qquad (*)$$

under the constraint $$\Psi(x)=\sum_{k=-j}^j \psi(x,k,j)=\sum_{k=-j}^j \frac{2 \pi}{j}\,e^{(i\,2\pi\,(k-1)\,x / j)}$$

Particularly inetresting also whether this equation $(*)$ comes in other notations such as in classical quantum mechanics. Please see this as a research type of question rather solving a mathematical problem.

I would really appreciat your help.

Thanks a lot

PS: although this question relates to Number theoretical function applied in physics? but it is not the same, here it is really about the differential equation.

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Welcome, to Physics.SE, al-Hwarizmi. As with most Stack Exchange site, we consider questions of the form "lets build a list of [something]" to be poor fits to the question and answer model we use here, and discourage them. –  dmckee May 30 '13 at 14:50
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closed as not constructive by dmckee May 30 '13 at 14:47

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