Evolution of the resonance wavefunction

Multiplying Eq. (11) by tjr (x,t) from the left and then subtracting from the result its complex conjugate, we are left with 

Comparing Eqs. (9) and (13), we can conclude that for sufficiently long times such that Eq. (9) is satisfied, we get that 

Using Eq. (3) and the fact that for sufficiently long times the behavior of the phase S(x,t) is portrayed in Eq. (7), we get that the real part of the local expectation value of the Hamiltonian reads 

Therefore, the local expectation value of the Hamiltonian reads 

Taking into consideration the observed analogy between the behavior of a wave packet at large times and a stationary state in the interaction region, we are prompt to make the following ansatz for the wavefunction f R (x, t) in the interaction region  

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