Two-dimensional hydrogen atom. I. Elliptic basis

A two-dimensional hydrogen atom is analyzed in elliptic coordinates. Separation of the variables reduces the problem to the solution of the Ince equation in the complex plane subject to certain boundary conditions. It is shown that in the limits $R\rightarrow 0$ and $R\rightarrow\infty$ ($R$ is a parameter that specifies the elliptic coordinates) the obtained solutions go over to polar and parabolic bases, respectively. The explicit form of the elliptic basis is given for the lowest quantum states.