# Partial differential equation in the plane

Hi everybody!

I would kindly ask you if somebody know some method to solve analitically the following equation (written in cylindrical coordinates):

$\Big[\frac{\partial^2}{\partial\rho^2}+\frac{1}{\rho}\frac{\partial}{\partial\rho}+\frac{\partial^2}{\partial z^2}-\frac{1}{\rho^2}+\alpha^2\big(\frac{1}{\sqrt{\rho^2+(z-d)^2}}-\frac{1}{\sqrt{\rho^2+(z+d)^2}}\big)^2\Big]P(\rho,z)=0$

This equation is a kind of Schrodinger equation indipendent of time: laplacian plus dipole potential squared (with charge situated at ρ=0,z=±d).

I wrote this equation in prolate spheroidal coordinates $(\rho=d\sqrt{(\mu^2-1)(1-\eta^2)},z=d\mu\eta)$:

$\big(\frac{1}{\sqrt{\rho^2+(z-d)^2}}-\frac{1}{\sqrt{\rho^2+(z+d)^2}}\big)^2=\big(\frac{\alpha}{d}(\frac{1}{\mu-\eta}-\frac{1}{\mu+\eta})\big)^2=\frac{\alpha^2}{d^2}\frac{4\eta^2}{(\mu^2-\eta^2)^2}$

$\Big[\frac{1}{d^2(\mu^2-\eta^2)}\Big((\mu^2-1)\frac{\partial^2}{\partial\mu^2}+2\mu\frac{\partial}{\partial\mu}-\frac{1}{\mu^2-1}-(\eta^2-1)\frac{\partial^2}{\partial\eta^2}-2\eta\frac{\partial}{\partial\eta}+\frac{1}{\eta^2-1}\Big)+\frac{\alpha^2}{d^2}\frac{4\eta^2}{(\mu^2-\eta^2)^2}\Big]P(\mu,\eta)=0$

and tried to solve it factorizing the behaviour at the singularities and expanding the regular part in series of μ,η arriving always at the contraddiction that all the coefficient must be zero.

I report the analysis of the singularities (when we look near a singularity we can discard the potential generated by the other charge because is not diverging):

$\Big[\frac{1}{d^2(\mu^2-\eta^2)}\Big((\mu^2-1)\frac{\partial^2}{\partial\mu^2}+2\mu\frac{\partial}{\partial\mu}-\frac{1}{\mu^2-1}-(\eta^2-1)\frac{\partial^2}{\partial\eta^2}-2\eta\frac{\partial}{\partial\eta}+\frac{1}{\eta^2-1}\Big)+\frac{\alpha^2}{d^2}\frac{1}{(\mu\pm\eta)^2}\Big]P(\mu,\eta)=0$

$P_1(\mu,\nu)=d\big(d(\mu\pm\eta)\big)^{-3/2+\nu}\sqrt{(\mu^2-1)(1-\eta^2)}$, $P_2(\mu,\nu)=d\big(d(\mu\pm\eta)\big)^{-3/2-\nu}\sqrt{(\mu^2-1)(1-\eta^2)}$.

I would need an analytic solution of this equation to give a solid support at a more general problem that I solved numerically that is to find the positive eigenvalues of this operator. I found that the value of the eigenvalue depend from the distance of the two charges: from a maximum values when the charges are very distant, it decrease bringing them near until it reach the value zero at a critical distance. The equation corresponds then to the zero mode of my differential operator and the solution will give, after imposing the boundary condition, an analytic formula for the critical distance.

I will be very glad for any advice about some methods or also simply for some reference where I can find some idea to solve this equation.

Thanks!

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