Y. Strauss
On the existence of Lyapounov variables for Schroedinger evolution
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ABSTRACT. The theory of (classical and) quantum mechanical microscopic irreversibility developed by B. Misra, I. Prigogine and M. Courbage
(MPC) and various other contributors is based on the central notion of a Lyapounov variable - i.e., a dynamical variable whose
value varies monotonically as time increases. Incompatibility between certain assumed properties of a Lyapounov variable and
semiboundedness of the spectrum of the Hamiltonian generating the quantum dynamics led MPC to formulate their theory in
Liouville space. In the present paper it is proved, in a constructive way, that a Lyapounov variable can be found within the standard
Hilbert space formulation of quantum mechanics and, hence, the MPC assumptions are more restrictive than necessary for the
construction of such a quantity. Moreover, as in the MPC theory, the existence of a Lyapounov variable implies the existence of
a transformation (the so called $\Lambda$-transformation) mapping the original quantum mechanical problem to an equivalent
irreversible representation. In addition, it is proved that in the irreversible representation there exists a natural time observable
splitting the Hilbert space at each $t>0$ into past and future subspaces.