Liouvillian eigenvalues

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

According to (1.22) and (1.26), the eigenvalues v of the Liouvillian L are distributed symmetrically around the point v = 0, and this implies that, even if the Hamiltonian H in physics is bounded from below, H > a 1, the Liouvillian L is as a rule unbounded. Except for this difference, practically all the Hilbert-space methods developed to solve the Hamiltonian eigenvalue problem in exact or approximate form may be applied also to the Liouvillian eigenvalue problem. In the time-dependent case, the L2 methods developed to solve the Schrodinger equation are now also applicable to solve the Liouville equation (1.7). 

In the computational technique of today, one may be able to handle secular problems of order n = 106 in the carrier space, and, since this implies m = 1012, it seems at first sight as if the Liouvillian eigenvalue... 

It should be emphasized, however, that the Liouvillian matrix L in the Hilbert-Schmidt approach is diagonalized as easily as the Hamiltonian matrix H, provided that one starts with the latter problem. For those who insist that they would prefer to solve the Liouvillian eigenvalue problem without any reference to the Hamiltonian—except for the definition (1.6)—this is, of course, a setback, but already the discussion above has shown that the ket-bra formalism offers certain advantages which should certainly not be neglected. 

Still the purpose of this article is to advocate that the time is now ripe to attack the Liouvillian eigenvalue problem LC = vC directly in terms of single-commutator methods and secular equations of the type (2.16). This approach should further be combined with ket-bra methods of the type developed in Section II in order to decompose the eigenelements C associated with degenerate eigenvalues v into components having the form of excitation operators of the type C = TfXTj. ... 

MSN. 138. T. Petrosky and I. Prigogine, Complex spectral representations and singular eigenvalue problem of Liouvillian in quantum scatering, in Symposium Quantum Physics and the Universe, Waseda University, Tokyo, Japan, 1992. 

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]... 

II. The Eigenvalue Problem for the Liouvillian in the Hilbert-Schmidt Theory 300... 

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

It should be observed, however, that in the literature many authors prefer to solve the eigenvalue problem (1.22) for the Liouvillian L directly in the operator space without any reference to the Hamiltonian formalism, to wave functions or ket-bra operators, and—in such a case—the algebraic conditions (1.32) are not necessarily satisfied for the approximate excitation operators D derived, and the study of these or similar conditions may become a special and sometimes crucial problem in the direct approach. ... 

A completely different approach in trying to calculate the eigenvalues of the Liouvillian is rendered by the equation-of-motion method, which was developed in nuclear physics12 13 and later introduced into quantum chemistry by several research groups.19 The basic idea is to try to solve the eigenvalue problem (1.22) by expanding the approximate eigenele-ment D in terms of a truncated basis B = Br of order m in the operator space, so that... 

Let us now also consider the eigenvalue problem (1.22) of the Liouvillian L. Forming the CBP with an arbitrary operator D, one obtains... 

One may utilize this fact to construct approximate solutions to the eigenvalue problem (1.22) for the Liouvillian. For this purpose, we will introduce a set B = B, Bi,. . . , Bm of m linearly independent HS operators in our operator space, and we will then try to expand the approximate eigenoperator D in this truncated basis, so that... 

At first sight, this result may seem rather uninteresting, since one of the purposes of the Liouvillian formalism is to try to solve the eigenvalue problem (1.22) directly in the operator space. On the other hand, it may be of some value if the approximate eigenoperator D = Bd obtained by solving (2.15) and (2.18) does not automatically satisfy the algebraic relations (1.32). In such a case, one may proceed by introducing an arbitrary normalized reference function associated with the reference operator... 

These results indicate that the double-commutator Liouvillian matrix has rather different properties compared to the ordinary matrix (2.64), and that the eigenvalues in the form of energy differences have a different form. It should also be observed that the solutions to the linear equation system (2.83) may have a rather different character.13... 

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... 

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