Periodic effective potentials for spin systems and new exact solutions of the one-dimensional Schr�dinger equation for the energy bands

Abstract

The technique developed in an earlier paper by the authors is used in conjunction with a representation of generalized coherent states to find new effective periodic potential fields that rigorously describe stationary states of (pseudo) spin systems of the type of a two-axis paramagnet in a magnetic field. The potentials depend strongly on several parameters, their profiles are rich in distinctive features of the type of double wells, two-hump barriers, fourfold minima and maxima, and in the bands interesting structural transformation take place (finite-gap property, band pairing, etc.). It is shown that the spin system corresponds to periodic and antiperiodic solutions with extremal energy levels in the 2S + 1 lowest bands (S is the spin). On the basis of the established spin-coordinate correspondence, new classes of exact solutions of the Schroedinger equation are found for energy bands with simple explicit expressions for the energy levels and wave functions for S = 0, 1/2, 1, 5/2, 3, 7/2, 4, 9/2, 5. The potentials are expressed in terms of elliptic functions and contain as special cases the finite-gap Lame-Ince potential and the Eckart and Morse potentials. Effective potentials are also constructed for Hamiltonians of the group SU(1,1).