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 infinite number of equidistant states with different energy spacings oi in each band is studied. Both bands are linearly dependent on time t. The interaction upsilon = (root omega,omega+/pi) tan pi s between the bands is considered to be equal for any pair of states from each band. Using the Fourier series transformation the instant eigenvalues E(t, s) are calculated which reveal the double periodicity in the energy-time plane. The corresponding eigenvalue surface in the (E, t, s)-space, apart from the triple periodicity, shows quite unexpected symmetry properties relative to the exchange of t and s, and relative to some inversions in the (E, t) plane. The latter one leads to a new equivalence between 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 diagonalization of H(t) for fixed t produces explicit expressions for the eigenvalues (adiabatic potential curves) and eigenstates (adiabatic basis). The time evolution operator is calculated both in the diabatic and adiabatic representations. The results are simplified for the special value of the interaction parameter.