Superoperator resolvent

Thus the matrix elements of the electron propagator are related to field operator products arising from the superoperator resolvent, El — H), that are evaluated with respect to N). In this sense, electron binding energies and DOs are properties of the reference state. 

The superoperator resolvent —is defined by the usual series expansion... 

However, this is only a cosmetic change, because the superoperator resolvent is an inverse operator and is only defined through its series expansion in Eq. (3.148). The way in which we can proceed, is to find a matrix representation of the superoperator resolvent. In order to do that we need a complete set of basis vectors like in the normal Hilbert space. However, the vectors in the superoperator formalism are operators and we therefore need a complete set of operators. Such a set of operators hn consists of a complete set of excitation and de-excitation operators with respect to the reference state This means that all other states of the system or all excited states of... 

The desired matrix representation of the superoperator resolvent is then obtained in two steps by the inner projection technique (Pickup and Goscinski, 1973), where the superoperator resolvent is projected in the space of the complete set of excitation and de-excitation operators hn - First, we insert the resolution of the superoperator identity twice in Eq. (3.149) leading to... 

To that purpose we start with the definition of the superoperator resolvent, i.e. 

This expression no longer contains the inverse of operators but the inverse of matrix representations of the operators. Prom comparison of Eq. (3.149) and Eq. (3.157) we may conclude that a matrix representation of the superoperator resolvent is given by... 

In (7.90) a slightly modified notation is introduced for convenience for the bra and ket vectors in the Liouville space for the resolvent superoperator... 

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]... 

It can be formally resolved using the superoperator adjoint to the unperturbed Fock operator ... 

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