a continuum in [@cotan]. This treatment needs to be modified for our case, but we first review it to set the basic technique in the context of the present up-down matrix formalism.

We thus consider a quantum mechanical system with a set of discrete states $|n \>$ and continuum states $|\a \ra$ where $\a$ denote the continuous labels of the state. The Hamiltonian is $H= H_0 + V$ where $H_0$ and $V$ are given by the following matrices: H\_0&=&(

[cc]{} E\_i\^i\_j & 0\
0 & E\_(-)

),\
V&=&(

[cc]{} 0 & V\^i\_\
V\^\_i & 0

). Note that $V$ has vanishing matrix elements between pairs of discrete or pairs of continuum states in agreement with the interactions $H_{\pm}$ of (\[heff\]). We look for the solution of the Schrödinger evolution equation $ i \frac{d}{dt} |\Psi(t) \ra  = H |\Psi(t) \ra$, with $|\Psi(t) \ra$ given by the formal vector: |(t) =(

[c]{} a\^i(t)e\^[-iE\_it]{}\
\^a(t)e\^[-iE\_at]{}

). and the initial conditions $a^m(0) = \d^m_n,~~ \phi^\a(0) =0$.

It is easy to see that the discrete and continuum components of $|\Psi(t) \ra$ satisfy: i [d dt]{} a\^n(t) &=& dV\^i\_i(E\_n -E\_)t  \^(t)\
i [d dt]{} \^(t) &=& \_[m]{} V\^\_m i(E\_-E\_m)t  a\^m(t) We integrate the second equation using 