Persistent real eigenvalues

It is evident that the spectra of H and W are essentially complex conjugate. For a persistent real eigenvalue E = E, one has... 

It is hence evident that this expectation value is going to be complex, even if is a persistent real eigenvalue. Let us now plot the function... 

In a similar way, one can plot the eigenvalues of the matrix H = ri 2T + ri 1 for any choice of basis 0 as functions of the parameter t] = pelx. Since the eigenvalues / are in general going to be complex, it is evidently necessary to carefully distinguish between the approximations to real persistent eigenvalues, to continuous eigenvalues in the complex plane, and to true physical resonances. 

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