Hermite equation, eigenvalue

The change of variable z = transforms equation (V-8) into a Hermite equation, the lowest eigenvalue of which is... 

In the q-coordinate system, the vibrational normal coordinates, the SA atom-dimensional Schrodinger equation can be separated into SA atom one-dimensional Schrodinger equations, which are just in the form of a standard harmonic oscillator, with the solutions being Hermite polynomials in the q-coordinates. The eigenvectors of the F G matrix are the (mass-weighted) vibrational normal coordinates, and the eigenvalues ( are related to the vibrational frequencies as shown in eq. (16.42) (analogous to eq. (13.31)). 

However, due to the lack of Hermiticity, the spectrum of H includes, in addition to the normal spectrum of the ordinary Hamiltonian H, the possibility of complex eigenvalues. The occurence of such then causes the general operator exp(-iHt/h) to be undefined unless one considers it in a projected subspace (42). Within such a subspace comprised of all physical solutions to the time dependent Schrodinger equation, equation (82) is perfectly well behaved. Let us consider again the time independent equation... 

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... 

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