1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 7
f (t2, T ) f (t1, T )
T T T1(t1) T3(t1) T2(t1)
Figure 1.9. Schematical form of radiation power difference at
times t1 and t2
Figure 1.10. Potential function U at time instants t1 and t2.
We now turn to a more general context. For simplicity of the heuristical exposi-
tion still sticking to a one-dimensional scenario, we start by considering a temporally
varying potential function U and set
f(t, ·) = −
U(t, ·), t ≥ 0.
We assume that U oscillates in time between the two extreme positions depicted
schematically in Figure 1.10.
In Figure 1.10 (l.), the potential well on left hand side is deeper than on the
right hand side, in Figure 1.10 (r.) the role of the deeper well has changed. As
t varies, we will observe a smoothly time dependent potential with two wells of
periodically and smoothly fluctuating relative depth. Just the function describing
the position of the deepest well will in general be discontinuous. It will play a
crucial role in the analysis now sketched.
We assume in the sequel for simplicity that U(t, x), t ≥ 0, x ∈ R, is a smooth
function such that for all t ≥ 0, U(t, ·) has exactly two minima, one at −1, the
other at 1, and that the two wells at −1 and 1 are separated by the saddle 0, where
U(t, 0) is assumed to take the value 0. Two moment pictures of the potential may
look as in Figure 1.10.
We further assume time periodicity for U, more formally that
U(t, ·) = U(t + 1, ·).