Superoperator eigenvalue problem

From (1.10), it follows that eigenvalues of the Hamiltonian superoperator correspond to poles of the Green s function, and therefore, to EADEs. Thus, we are faced with an eigenvalue problem [33,34],... 

Let us now consider the eigenvalue problem for the Liouvillian superoperator L in the form... 

The key problem in the Liouvillian formalism is not only the direct solution of the eigenvalue problem (1.22) in the operator space but also— in the case of degenerate eigenvalues v—the separation of the eigenele-ments into components having the form of excitation operators of the special type (1.23) associated with specific initial and final states. Letting the superoperator G work on (1.22), one obtains... 

It is evident that, as long as the metric superoperator G satisfies the conditions (3.3), this is a legitimate approach for solving the original eigenvalue problem (1.22). 

The classical Liouvillian operator Zc, which is the classical limit of the Landau-von Neumann superoperator in Wigner representation, can also be analyzed in terms of a spectral decomposition, such as to obtain its eigenvalues or resonances. Recent works have been devoted to this problem that show that the classical Liouvillian resonances can be obtained as the zeros of another kind of zeta function, which is of classical type. The resolvent of the classical Liouvillian can then be obtained as [60, 61]... 

F.6.4.2. Lineshape Models. The Mossbauer lineshape can be influenced by all relaxation modes of the Fokker-Planck equation (see Section D.3). Because the relative importance of these modes depends on their population, it should be necessary to know both the eigenvalues of Brown s equation and the amplitudes of the associated modes. In fact, to determine the lineshape, it is necessary to connect the dynamics of the stochastic vector m given by Brown s equation with the quantum dynamics of the nuclear spin. This necessitates the use of superoperator Fokker-Planck equations and, to our knowledge, the problem has not yet been completely solved. 

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