The Complex-Energy Effective Hamiltonian

For a time-independent H, the eigenstates, ipj, with energy, Ej, that satisfy the time-dependent Schrodinger equation are 

So the first thing we get by allowing H to have complex diagonal elements is that state populations (diagonal elements of the density matrix) can decay exponentially with the expected decay rate, (1 /tj = r /fi), and this exponential decay corresponds to a Lorentzian broadened level (as opposed to a (5-function) centered at e3 with the expected full width at half maximum 

The usual orthonormality and completeness relationships must be replaced by 

Nondegenerate perturbation theory enables algebraically transparent analysis of the interactions between any number of decaying quasi-eigenstates pro- 

These results are very similar to the normal two-level problem, except (i) the real parts of the energies exhibit level repulsion, but this repulsion is reduced (but never reversed) by the 5r2/2 5e term in the denominator of Eqs. (9.3.12a) and (9.3.12b) (h) the imaginary parts of the energies exhibit level attraction (the interaction causes the widths of the mixed states to become more similar), and this attraction is reduced (but never reversed) by the 48e2/8Y term in the denominator of Eqs. (9.3.12a) and (9.3.12b) (Hi) when 5T — 0, the level widths are unaffected by the interaction (iv) when Se = 0 but 5Tj V, the level positions are unaffected by the interaction (because the narrower level is symmetrically surrounded by the much broader level and the usual level shift by V is suppressed). This amounts to an intuitively-sensible extension of the normal (real E, Hermitian H) two-level problem. 

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