The Schrödinger equation is
$$i \hbar \frac{\partial}{\partial t} \Psi(x, t) =- \frac{\hbar^2}{2m}\Delta \Psi(x,t)+V(x)\Psi(x, t)$$
If $\Psi(x,t) = e^{-iEt}u(x)$, we will get a ground state equation. For this case, it is $S^1$ symmetric. But as I know, in gravitational field , the orbital can be a elliptic, so it means the Schrödinger equation should have elliptic symmetric solution, is it ?
What is elliptic symmetric: for the Schrödinger equation, it is an elliptic symmetric solution, if it has form solution $\Psi(x,t) = (a\cos Et+ i b\sin Et)u(x)$, where $a,b>0$ is constant.