Extending a solution for $\left(H_\text{osc} + \delta^{(3)}(\vec r) \frac{\partial}{\partial r} r \right) \Psi(\vec r)$ to Gaussian potentials

Question

I am a undergrad and currently trying to understand and use the following paper:

Th. Busch u. a. Two Cold Atoms in a Harmonic Trap. 1997. URL: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.9047.

There, the authors derive an analytical solution to $$ \left(H_\text{osc} - E + \sqrt 2 \pi a_0 \delta^{(3)}(\vec r) \frac{\partial}{\partial r} r \right) \Psi(\vec r) = 0, $$ which models the center-of-mass part of two interacting atoms in a harmonic trapping potential. Their interaction is the regularized $\delta$-function $$ \delta^{(3)}(\vec r) \frac{\partial}{\partial r} r. $$

The solution they give is $$ \Psi(\vec r) = \frac 12 \pi^{-3/2} A \exp\left(-\frac{r^2}2\right) \Gamma(-\nu) U\left(-\nu, \frac 32, r^2 \right),$$ where $A$ is for normalization and $U$ is the confluent hypergeometric function.

With Mathematica, I have verified that this $\Psi(\vec r)$ is indeed a solution of the Schrödinger equation.

In section 3 of the paper, they sketch how this can be done in one dimension as well. I have not succeeded in deriving $\Psi(x)$ for that case yet, but I hope that this will come eventually.

My goal is to find an analytical solution to a one dimensional anharmonic oscillator that is pertubed with a gaussian potential.

Since I am currently lacking $\Psi(x)$ for the one dimensional $\delta$-pertubed oscillator, I wanted to try the three dimensional case first. In electrodynamics we had solved potential problems with a Green's function and used convolution to extend it. I think that the same might be possible here. Convolving the Gauss potential with the unregularized $\delta^{(3)}$-function gave me $\sqrt{2} \pi \mathrm e^{-r^2} (2 r^2-1)$.

Then I did a convolution of the 3D-$\Psi(\vec r)$, but that does not solve the adapted Schrödinger equation, as far as I can tell.

Is that even a legitimate way to extend the solution in the paper to other potentials? If so, what would be the right way to do this?

Answer 1

Right now you have the equation $L\Psi = 0$ where $L$ is the linear operator in parentheses in your first equation.

The situation to use green's functions is when you have a solution for $\Psi$ of $L\Psi = \delta(x-x')$, and you want to find a solution for $\Psi$ of $L\Psi = f(x)$ for some arbitrary function $f(x)$.

The situation you have is different. Here it is not the right hand side that is changing$-$your right hand side is always zero$-$but it is the operator that is changing. So the method of green's functions is not applicable here.

As far as ways to solve the schroedinger equation, I think it is a hard problem. I want to say that most people just do it numerically. Perhaps some one else will have a better answer.