Yn. Demkov et al., DOUBLY PERIODICAL IN TIME AND ENERGY EXACTLY SOLUBLE SYSTEM WITH 2 INTERACTING SYSTEMS OF STATES, Journal of physics. A, mathematical and general, 28(15), 1995, pp. 4361-4380
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.