Library of Congress Cataloging in Publication DataSolvable models in quantum mechanics.(Texts and monographs in physics) Bibliography: p. Includes index.1. Quantum theory-Mathematical models.
Library of Congress Cataloging in Publication DataSolvable models in quantum mechanics.(Texts and monographs in physics) Bibliography: p. Includes index.1. Quantum theory-Mathematical models.
Perron-Frobenius type results are proved for discrete, Markovian, quantum stochastic processes.
IntroductionAt the beginning of the century, Perron [19] and Frobenius [9,10] discovered many important spectral properties possessed by matrices with positive entries. There now exists a vast literature extending some of their results to positive operators on a large class of ordered vector spaces, the most successful results being with compact operators and/or cones with a lattice ordering or a large interior. We refer the reader to [11,15,22,32] and the references quoted therein. Here we regard the original Perron-Frobenius theory as being concerned with the spectral properties of positive operators on finite dimensional, commutative C*-algebras, and give a noncommutative version of this theory, at least for finite-dimensional C*-algebras.The first part of the Perron-Frobenius story tells us that the spectral radius of a positive matrix with positive entries is an eigenvalue, possessing a positive eigenvector. Moreover, if the matrix is irreducible in a certain sense, then the spectral radius is a simple eigenvalue, and (apart from scalar multiplication) there are no other positive eigenvectors. It is this part of the theory which has received most attention by other authors, referred to above. In §2 we give our generalization of this to finite-dimensional C*-algebras.Perron and Frobenius also showed that the spectrum and eigenvectors of an irreducible positive matrix had certain multiplicative properties. This part of the theory has not received nearly as much attention, although Rota [20] and Schaefer [21] obtained some results in this direction for certain lattice ordered spaces, namely IP-spaces and commutative C*-algebras. Analogous results were obtained by St0rmer for ergodic groups of automorphisms on von Neumann algebras [29]. In §3 and §4 we study multiplicative properties associated with the spectrum of an irreducible positive operator on a finite-dimensional C*-algebra. In §3, it is mainly the Jordan structure which is important, but the C*-structure takes over in §4 when we restrict attention to those maps which satisfy the Schwarz inequality which we call Schwarz maps.In [4] and [5] Davies has proposed concepts of recurrence and transience for certain continuous-time Markovian quantum stochastic processes. In §3 we also propose definitions of recurrence and transience for discrete-time Markovian quantum stochastic processes, which are different from those of Davies. (Davies's ideas easily carry over from continuous time to discrete time.) Discrete non-Markovian quantum stochastic processes have been studied recently by Accardi [1,2] and Lindblad [16].
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