Schrodinger equation in two dimension

[santale]

Hi everybody!

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

[tex]\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[/tex]

This equation is a kind of Schrodinger equation indipendent of time: laplacian plus dipole potential squared (with charge situated at [itex]\rho=0,z=\pm d[/itex]).

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

[tex]\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}[/tex]

[tex]\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 [/tex]

and tried to solve it factorizing the behaviour at the singularities and expanding the regular part in series of [itex]\mu,\eta[/itex] 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):

[tex]\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[/tex]

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

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!

 

[jvicens]

Hello, thank you for your question. This is a very interesting and complex problem. Unfortunately, I do not have an analytic solution for this particular equation. However, I can suggest some possible approaches that you can try.

1. Method of separation of variables: This method involves assuming a separable solution of the form P(\rho,z) = R(\rho)Z(z). Substituting this into the equation, you can reduce it to two ordinary differential equations, one for R(\rho) and one for Z(z). Solving these equations separately may lead to an analytic solution.

2. Perturbation theory: Since you mentioned that you have solved the problem numerically, you can use the numerical solution as a starting point and apply perturbation theory to find an analytical approximation. This method involves expanding the solution in a power series and solving for the coefficients.

3. Integral transforms: This method involves transforming the equation into a different form using integral transforms such as Fourier transform or Laplace transform. This can sometimes simplify the equation and make it easier to solve analytically.

I hope these suggestions are helpful to you. You can also consult with other experts in the field or refer to research papers on similar problems for more insights. Good luck with your research!