DOUBLY PERIODICAL IN TIME AND ENERGY EXACTLY SOLUBLE SYSTEM WITH 2 INTERACTING SYSTEMS OF STATES

The time-dependent matrix Schrodinger equation ic/l partial derivative l/partial derivative psi = H(t)psi, describing two bands of an infini te number of equidistant states with different energy spacings oi in e ach band is studied. Both bands are linearly dependent on time t. The interaction upsilon = (root omega,omega+/pi) tan pi s between the band s is considered to be equal for any pair of states from each band. Usi ng the Fourier series transformation the instant eigenvalues E(t, s) a re calculated which reveal the double periodicity in the energy-time p lane. The corresponding eigenvalue surface in the (E, t, s)-space, apa rt from the triple periodicity, shows quite unexpected symmetry proper ties relative to the exchange of t and s, and relative to some inversi ons in the (E, t) plane. The latter one leads to a new equivalence bet ween weak and strong coupling, a new kind of pseudocrossing and a new concept of antidiabatic states. The Fourier transformation reduces the problem to a 2 x 2 first-order differential operator. The diagonaliza tion of H(t) for fixed t produces explicit expressions for the eigenva lues (adiabatic potential curves) and eigenstates (adiabatic basis). T he time evolution operator is calculated both in the diabatic and adia batic representations. The results are simplified for the special valu e of the interaction parameter.