Quantum mechanics with coordinate-dependent mass

We study a motion of quantum particles, whose properties depend on one coordinate so that they can move freely in the perpendicular direction. A rotationally symmetric Hamiltonian is derived and applied to study a general interface formed between two semiconductors. We predict the existence of peculiar electron states localized at the interface which do not occur within the framework of conventional quantum mechanics. They appear whenever the two bulk dispersions intersect. These shallow states lie near the point of intersection and are restricted to a finite range of perpendicular momentum. The scattering of carriers by the interface is discussed.