Periodic effective potentials for spin systems and new exact solutions of the one-dimensional Schr\"odinger equation for the energy bands

O. B. Zaslavskii ; V. V. Ulyanov

Teoretičeskaâ i matematičeskaâ fizika, Tome 71 (1987) no. 2, pp. 260-271

Résumé

Using the technique developed in the previous paper by the authors and based on the representation of generalized coherent states, new effective periodic potential fields are found which describe rigorously stationary states of (pseudo) spin systems of the type of two-axis paramagnet in a magnetic field. The potentials change considerably depending on several parameters, in their profiles some peculiar shapes abound, of the type of double wells, two-hump barriers, quartuc minima or maxima, and interesting phenomena take place in the zones (finite-zoneness, pairing of zones, etc.). It is shown that spin systems are connected with (anti) coherent states with extremal energy levels in $2S +1$ lower zones ($S$ being the spin). On the basis of the spin-coordinate correspondence obtained, new classes of exact solutions of the Schrödinger equation are found for the energy zones with simple explicit expressions for energy levels and wave functions at $S=0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5$. The potentials are expressed in terms of elliptic functions and include, as particular cases, the finite-zone Lame–Eins potential, Eckart and Morse potentials. Effective potentials for the Hamiltonians of the $SU(1,1)$ group are also constructed.