Planar hydrogen-like atom in inhomogeneous magnetic fields: Exactly or quasi-exactly solvable models

We use a simple mathematical method to solve the problem of a two-dimensional hydrogen-like atom in the inhomogeneous magnetic fields $\mathbf B=(k/r)\mathbf z$ and $\mathbf B=(k/r^3)\mathbf z$. We construct a Hamiltonian that takes the same form as the Hamiltonian of a hydrogen-like atom in the homogeneous magnetic fields and obtain the energy spectrum by comparing the Hamiltonians. The results show that the whole spectrum of the atom in the magnetic field $\mathbf B=(k/r)\mathbf z$ can be obtained, and the problem is exactly solvable in this case. We find analytic solutions of the Schrödinger equation for the atom in the magnetic field $\mathbf B=(k/r^3)\mathbf z$ for particular values of the magnetic strength $k$ and thus present a quasi-exactly solvable model.