Perturbation of a negative potential time dependent problem

Hello,
  I am trying to calculate the probability that the n = 3 state will be excited at t = infinity for the case where b<<< a. and to show that for any value of b only the odd n states will be excited at t = infinity.  

my initial conditions and setup are:

V(x,t) = Vinit(x) exp(-lambda times t) where I have a negative potential in the center of an infinite square well for the case where the perturbation fades away exponentially in time.  

I set my potential at -V in the center between the -a and a boundaries and the perturbation extends from -b to +b in the center so the pertubation is as follows:

  Vinit(x) = 0 for -a < or = x , -b
  Vinit(x) = -V for -b, or = x , or = b
  Vinit(x) = 0 for = b, x, or = +a

the sq well is like this:  |                         |
                                  |                         |
                                  |_____       _____|
                               -a      -b |     | b      a
                                            |___|

as best as I can draw it in text chars.  

and my unperturbed wavefunctions between a and -a are:  

    square root of 1/a sin(n*pi*x/2*a) for n even, and same only cos for n odd

I hope this gives enough detail for you.  I have been working on this one but am having trouble with the t = infinity and b <<< a conditions.