We describe a method for the calculation of guaranteed bounds for the K lowest eigenvalues of second-order problems with Neumann boundary conditions. Using P 2 approximations for the eigenfunctions and RT 1 approximations for the gradients of the eigenfunctions in H(div; , an error bound for the eigenfunctions is established for weak approximations in H 1(\Omega\Gamma . In addition, the rest of the spectrum will be bounded by a domain decomposition method; step by step the eigenvalue problem is decomposed into simpler geometrical situations, where sufficient information on the spectrum is available. 1.

We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form x = A(t)x+f(t) (), with f having precompact range, which will be then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonnant case where e isp(f) may intersect the spectrum of the monodromy operator P of () (here sp(f) denotes the Carleman spectrum of f ). We show that if () has a bounded uniformly continuous mild solution u and oe \Gamma (P )ne isp(f) is closed, where oe \Gamma (P ) denotes the part of oe(P ) on the unit circle, then () has a bounded uniformly continuous mild solution w such that e isp(w) = e isp(f) . Moreover, w is a "spectral component" of u. This allows to solve the general Massera-typed problem for almost periodic solutions. Various spectral criteria for the existence of almost periodic, quasi-periodic mild solutions to () are ...

This paper shows how C-numerical-range related new strucures may arise from practical problems in quantum control—and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function |tr{C † UAU †}|. In quantum control of n qubits one may be interested (i) in having U ∈ SU(2 n) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e. to the n-fold tensor product SU(2) ⊗ SU(2) ⊗ · · · ⊗ SU(2). Interestingly, the latter then leads to a novel entity, the local C-numerical range Wloc(C, A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying paper on Relative C-Numerical Ranges for Application in Quantum Control and Quantum Information [1]. We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn’s

Abstract. We study the quantization of the regularized hamiltonian, H, of the compactified D = 11 supermembrane with nontrivial winding. By showing that H is a relatively small perturbation of the bosonic hamiltonian, we construct a Dyson-type series for the heat kernel of H and prove its convergence in the topology of the von Neumann-Schatten classes so that e −Ht is ensured to be of finite trace. The results provided have a natural interpretation in terms of the quantum mechanical model associated to regularizations of compactified supermembranes. In this direction, we discuss the validity of the Feynman path integral description of the heat kernel for D = 11 supermembranes and obtain a matrix Feynman-Kac formula. 1.

We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic.

Abstract: We introduce a non-commutative extension of Tsirelson-Vershik’s noises [TV98, Tsi04], called (non-commutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, in particular, capable of producing Arveson’s product system of type I and type II [Arv03]. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalar-expected) continuous Bernoulli shift is either finite or of type III. The role of (‘classical’) G-stationary flows for Tsirelson-Vershik’s noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Lévy processes. They lead, in particular, to units and ‘logarithms ’ of units in Arveson’s product systems [Kös04a]. Furthermore, we introduce (non-commutative) white noises, which are operator algebraic versions of Tsirelson’s ‘classical ’ noises. We give examples coming from probability, quantum probability and from Voiculescu’s theory of free probability [VDN92]. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: non-commutative extensions of stochastic Itô integration, stochastic logarithms and exponentials.

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The theory of one-parameter semigroups provides a good entry into the study of the properties of non-self-adjoint operators and of the evolution equations associated with them. There are many situations in which such an operator A arises by linearizing some non-linear evolution equation around a stationary point. The stability

A principal result of the paper is that if E is a symmetric Banach function space on the positive half-line with the Fatou property then, for all semi-finite von Neumann algebras (M; ), the absolute value mapping is Lipschitz continuous on the associated symmetric operator space E(M; ) with Lipschitz constant depending only on E if and only if E has non-trivial Boyd indices. It follows that if

We consider a quantum system in contact with a heat bath consisting in an infinite chain of identical sub-systems at thermal equilibrium at inverse temperature β. The time evolution is discrete and such that over each time step of duration τ, the reference system is coupled to one new element of the chain only, by means of an interaction of strength λ. We consider three asymptotic regimes of the parameters λ and τ for which the effective evolution of observables on the small system becomes continuous over suitable macroscopic time scales T and whose generator can be computed: the weak coupling limit regime λ → 0, τ = 1, the regime τ → 0, λ 2 τ → 0 and the critical case λ 2 τ = 1, τ → 0. The first two regimes are perturbative in nature and the effective generators they determine is such that a non-trivial invariant sub-algebra of observables naturally emerges. The third asymptotic regime goes beyond the perturbative regime and provides an effective dynamics governed by a general Lindblad generator naturally constructed from the interaction Hamiltonian. Conversely, this result shows that one can attach to any Lindblad generator a repeated quantum interactions model whose asymptotic effective evolution is generated by this Lindblad operator. AMS classification numbers: 81Q99