I have a function $\psi(x,y)$ which I have found by numerically minimising the energy of a Hamiltonian.
The problem is that there are multiple functions (let's say $2$ in this case for the sake of simplicity) that have the same energy. The way the code works, this $\psi$ is the sum of all degenerate functions: $$ \psi(x,y) = \phi_1(x,y) + \phi_2(x,y).$$
My question is: can I somehow extract $\phi_1$ and/or $\phi_2$ from $\psi$ purely form mathematical manipulation? I.e. not through perturbation theory.
I can guess some relationship between the two functions: e.g. that one if the $\pi/2$ rotation of the other. In which case I can rewrite this as: $$ \psi(\mathbf{r}) = \phi_1(\mathbf{r})+\phi_1(R^{\pi/2}\mathbf{r}).$$
Can I solve this particular case to find $\phi_1$?