Random repeated interaction quantum systems

We consider a quantum system S interacting sequentially with independent systems E_m, m=1,2,... Before interacting, each E_m is in a possibly random state, and each interaction is characterized by an interaction time and an interaction operator, both possibly random. We prove that any initial state converges to an asymptotic state almost surely in the ergodic mean, provided the couplings satisfy a mild effectiveness condition. We analyze the macroscopic properties of the asymptotic state and show that it satisfies a second law of thermodynamics. We solve exactly a model in which S and all the E_m are spins: we find the exact asymptotic state, in case the interaction time, the temperature, and the excitation energies of the E_m vary randomly. We analyze a model in which S is a spin and the E_m are thermal fermion baths and obtain the asymptotic state by rigorous perturbation theory, for random interaction times varying slightly around a fixed mean, and for small values of a coupling constant.Comment: Statements of Theorem 1.5 and 3.2, and proof of Theorem 3.3 modified. To appear in Comm. Math. Phy

Measurable Choice and the Invariant Subspace Problem

In [1], J. Dyer, A. Pedersen and P. Porcelli announced that an affirmative answer to the invariant subspace problem would imply that every reductive operator is normal. Their argument, outlined in [1], provides a striking application of direct integral theory. Moreover, this method leads to a general decomposition theory for reductive algebras which in turn illuminates the close relationship between the transitive and reductive algebra problems. The main purpose of the present note is to provide a short proof of the technical portion of [1] : that invariant subspaces for the direct integrands of a decomposable operator can be assembled in a measurable fashion . The general decomposition theory alluded to above will be developed elsewhere in a joint work with C. K. Fong, though we do present a summary of some of its consequences below

Relatively Transitive-operator Algebras.

PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/180301/2/7306779.pd

Spectral multiplicity for tensor products of normal operators

Two normal operators N 1 {N_1} and N 2 {N_2} are constructed such that for any pair m 1 {m_1} and m 2 {m_2} of their respective multiplicity functions, the ’convolution’ ( m 1 ∗ m 2 ) ( λ ) ≡ Σ { m 1 ( λ 1 ) ⋅ m 2 ( λ 2 ) | λ 1 ⋅ λ 2 = λ } ({m_1} * {m_2})(\lambda ) \equiv \Sigma \{ {m_1}({\lambda _1}) \cdot {m_2}({\lambda _2})|{\lambda _1} \cdot {\lambda _2} = \lambda \} fails to be a multiplicity function for the tensor product N 1 ⊗ N 2 {N_1} \otimes {N_2} .</p

Compact Operators in Reductive Algebras

Let be a Hilbert space and denote the collection of (bounded, linear) operators on by . Throughout this paper, the term ‘algebra’ will refer to a subalgebra of ; unless otherwise stated, it will not be assumed to contain I or to be closed in any topology.An algebra is said to be transitive if it has no non-trivial invariant subspaces. The following lemma has revolutionized the study of transitive algebras. For a pr∞f and a general discussion of its implications, the reader is referred to [5].</jats:p

Test problems for operator algebras

Kaplansky’s test problems, originally formulated for abelian groups, concern the relationship between isomorphism and direct sums. They provide a "reality check" for purported structure theories. The present paper answers Kaplansky’s problems in operator algebraic contexts including unitary equivalence of von Neumann algebras and equivalence of representations of (non self-adjoint) matrix algebras. In particular, it is shown that matrix algebras admitting similar ampliations are themselves similar.</p

ON THE STRUCTURE OF SUPPORT POINT SETS

ABSTRACT. Let X be a metrizable compact convex subset of a locally convex space. Using Choquet’s Theorem, wc determine the structure of the support point set of X when X has countably many extreme points. We also characterize the support points of certain families of analytic functions