quantum mechanics - What is known about the hydrogen atom in $d$ spatial dimensions?

In a first (or second) course on quantum mechanics, everyone learns how to solve the time-independent Schrödinger equation for the energy eigenstates of the hydrogen atom: $$ \left(-\frac{\hbar^2}{2\mu} \nabla^2 - \frac{e^2}{4\pi\epsilon_0 r} \right)|\psi\rangle = E|\psi\rangle .$$ The usual procedure is to perform separation of variables to obtain a radial equation and an angular equation, which are solved separately for the radial wavefunction $R_{n\ell}(r)$ and the spherical harmonics $Y_\ell^m(\theta, \phi)$. By combining these factors, we obtain the hydrogenic stationary-state wavefunctions $$ \psi_{n\ell m}(r, \theta, \phi) = R_{n\ell}(r) Y_\ell^m(\theta, \phi). $$ It seems to me that nothing about this problem (which is essentially the quantum-mechanical analogue of the classical two-body problem) is inherently three-dimensional, so we can just as well consider the corresponding problem in $d = 1, 2, 4, 5, \dots$ spatial dimensions. In fact, by transforming into hyperspherical coordinates \begin{align} x_1 &= r \cos(\phi_1) \\ x_2 &= r \sin(\phi_1) \cos(\phi_2) \\ x_3 &= r \sin(\phi_1) \sin(\phi_2) \cos(\phi_3) \\ &\vdots\\ x_{d-1} &= r \sin(\phi_1) \cdots \sin(\phi_{d-2}) \cos(\phi_{d-1}) \\ x_d &= r \sin(\phi_1) \cdots \sin(\phi_{d-2}) \sin(\phi_{d-1}) \,. \end{align} the same procedure employed in the 3D case should carry over (with a considerable increase in algebraic complexity). Has this problem been solved before? If so, what is known about the solution? In particular,

• How many quantum numbers are required to describe hydrogenic stationary states in $d$ spatial dimensions? Do these quantum numbers have a clear physical interpretation, like $n, \ell, m$?


• What is the $n$-dimensional analogue of the Bohr formula $E_n = E_1/n^2$? Do energies continue to depend on a single (principal) quantum number? As the dimension tends to infinity, is the energy spectrum discrete or continuous?


• What is the $d$-dimensional analogue of the spherical harmonics $Y_\ell^m(\theta, \phi)$? Can these functions be described as eigenfunctions of $d$-dimensional angular momentum operators, analogous to $L^2$ and $L_z$? If so, what are the eigenvalues?



• Is there a reasonable closed form for the $d$-dimensional hydrogenic stationary-state wavefunctions? If not, is there a reasonable (asymptotic) approximation formula?