Stationary solutions with outgoing waves

Consider the one dimensional TISE in Eq. (24), where we allow x to vary between -L and L, i.e., in the interaction region only. In order to solve this equation, we must supplement it with some boundary conditions. Although motivated from different quantum phenomena, Siegert [30] was the first to introduce the idea of solving the TISE with outgoing BCs, also known as Siegert boundary conditions or radiation boundary conditions. In one dimension, these outgoing BCs read 

In general, the eigenvalues and eigenstates are complex. It is common to divide the spectrum of the Hamiltonian with Siegert boundary conditions into four parts  

If k is purely imaginary and positive, then these states correspond to bound states with asymptotic behavior p( x L) xr . The bound state solutions are the only solutions with positive imaginary values of the wave vector [31]. 

The second type of solutions are those for which k is purely imaginary and negative. These states are called antibound states and have the asymptotic behavior of 4 ( x L) a e+ww. 

The next type of states and those that will interest us the most are the resonance states for which k = kT - ikir where kr and k are real positive numbers. These are outgoing states because the real part of k is positive. As will be shown below, these states diverge asymptotically. 

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