Liouville operator, eigenvalues

The formal similarity between Eq. (10) and the time-dependent Schrodinger equation is striking, and we shall indeed develop methods which are very reminiscent of quantum mechanics. In particular, we may calculate the eigenfunctions and eigenvalues of the unperturbed Liouville operator L0. We look for solutions of ... 

Given that the total hamiltonian may be written as Hw = P2/2M + hw(R), the adiabatic eigenfunctions a R) are the solutions of the eigenvalue problem, hw(R) ot R) = Ea(R) a R). In this adiabatic basis the quantum-classical Liouville operator has matrix elements [12],... 

An important aspect of this observation is that similar constant correlation lengths are also obtained classically.67 In the classical case, however, the correlation lengths attain a constant value only after a short relaxation time. The very existence of classical relaxation is the major difference with respect to the quantum behavior. It results, as discussed in previous sections, from the continuous spectrum for this classical system the classical cross correlation functions are not identically zero. Rather we conjecture that they rapidly decay to zero as k — k increases, where k and k are eigenvalues of the classical Liouville operator. In contrast, the quantum cross correlation functions are essentially zero whenever (the discrete) n — m 0. One further... 

A simple but nonrigorous version of this proof is the following. Since the Liouville operator is Hermitian its eigenvalues A are real and its eigenfunctions x(r) are orthogonal where... 

Properties (ii)-(iv) of the classical Liouville equation are a bit troublesome. The Hermiticity of the classical Liouville operator (f ) implies that its eigenvalues are real. Thus, p(X, t) must exhibit oscillatory temporal behavior and appears not to decay to a unique stationary state in the hmit t oo. This raises the question of how do we describe the irreversible decay of a system to a imique equilibrium state. The time-reversal invariance of... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

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