LIOUVILLE EXTENSION OF QUANTUM-MECHANICS - ONE-DIMENSIONAL GAS WITH DELTA-FUNCTION INTERACTION

A one-dimensional quantum gas with delta-function interaction is known to be integrable for any finite number of particles. We show that in the thermodynamic limit invariants of motion are destroyed in conjunct ion with a class of singular density matrices that appear commonly in both equilibrium and nonequilibrium statistical mechanics. The system becomes nonintegrable, due to Poincare resonances; However, the eigenv alue problem of the Liouville-von Neumann operator L-H may be solved i n an extended functional space that contains the singular density matr ices. We may in this way describe the approach to equilibrium. This is a simple example of the Liouville space extension of quantum mechanic s, due to Prigogine and his co-workers. The fundamental quantity in th is formulation is the probability (and not the wave function). We obta in indeed an irreducible complex spectral representation of L-H. Irred ucible means that the eigenstates of L-H cannot be implemented by wave functions, and complex means that the eigenvalues are complex numbers , whose imaginary parts determine relaxation times of the system. The dynamical evolution is described by a semigroup. Due to dissipative ef fects the real part of the eigenvalues is no more a difference of ener gy levels, as is the case in standard quantum mechanics. [S1050-2947(9 7)07311-3].