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

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We use a simple mathematical method to solve the problem of a two-dimensional hydrogen-like atom in the inhomogeneous magnetic fields B = (k/r)z and B = (k/r3)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 B = (k/r)z can be obtained, and the problem is exactly solvable in this case. We find analytic solutions of the Schrodinger equation for the atom in the magnetic field B = (k/r3)z for particular values of the magnetic strength k and thus present a quasi-exactly solvable model.