3,287 research outputs found

    Schr\"odinger operator on homogeneous metric trees: spectrum in gaps

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    The paper studies the spectral properties of the Schr\"odinger operator AgV=A0+gVA_{gV} = A_0 + gV on a homogeneous rooted metric tree, with a decaying real-valued potential VV and a coupling constant g≥0g\ge 0. The spectrum of the free Laplacian A0=−ΔA_0 = -\Delta has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation gVgV gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of gg if the potential VV has a fixed sign. Assuming that the latter condition is satisfied and that VV is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit g→∞g\to\infty. Depending on the sign and decay of VV, this asymptotics is either of the Weyl type or is completely determined by the behaviour of VV at infinity.Comment: AMS LaTex file, 47 page

    Process membership in asynchronous environments

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    The development of reliable distributed software is simplified by the ability to assume a fail-stop failure model. The emulation of such a model in an asynchronous distributed environment is discussed. The solution proposed, called Strong-GMP, can be supported through a highly efficient protocol, and was implemented as part of a distributed systems software project at Cornell University. The precise definition of the problem, the protocol, correctness proofs, and an analysis of costs are addressed

    Stability of the magnetic Schr\"odinger operator in a waveguide

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    The spectrum of the Schr\"odinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schr\"odinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own

    Exact Casimir Interaction Between Semitransparent Spheres and Cylinders

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    A multiple scattering formulation is used to calculate the force, arising from fluctuating scalar fields, between distinct bodies described by δ\delta-function potentials, so-called semitransparent bodies. (In the limit of strong coupling, a semitransparent boundary becomes a Dirichlet one.) We obtain expressions for the Casimir energies between disjoint parallel semitransparent cylinders and between disjoint semitransparent spheres. In the limit of weak coupling, we derive power series expansions for the energy, which can be exactly summed, so that explicit, very simple, closed-form expressions are obtained in both cases. The proximity force theorem holds when the objects are almost touching, but is subject to large corrections as the bodies are moved further apart.Comment: 5 pages, 4 eps figures; expanded discussion of previous work and additional references added, minor typos correcte

    Strong contraction of the representations of the three dimensional Lie algebras

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    For any Inonu-Wigner contraction of a three dimensional Lie algebra we construct the corresponding contractions of representations. Our method is quite canonical in the sense that in all cases we deal with realizations of the representations on some spaces of functions; we contract the differential operators on those spaces along with the representation spaces themselves by taking certain pointwise limit of functions. We call such contractions strong contractions. We show that this pointwise limit gives rise to a direct limit space. Many of these contractions are new and in other examples we give a different proof

    Condition for equivalence of q-deformed and anharmonic oscillators

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    The equivalence between the q-deformed harmonic oscillator and a specific anharmonic oscillator model, by which some new insight into the problem of the physical meaning of the parameter q can be attained, are discussed

    Maslov index, Lagrangians, Mapping Class Groups and TQFT

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    Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface.Comment: 31 pages, 11 Figures. to appear in Forum Mathematicu

    Fisheye Consistency: Keeping Data in Synch in a Georeplicated World

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    Over the last thirty years, numerous consistency conditions for replicated data have been proposed and implemented. Popular examples of such conditions include linearizability (or atomicity), sequential consistency, causal consistency, and eventual consistency. These consistency conditions are usually defined independently from the computing entities (nodes) that manipulate the replicated data; i.e., they do not take into account how computing entities might be linked to one another, or geographically distributed. To address this lack, as a first contribution, this paper introduces the notion of proximity graph between computing nodes. If two nodes are connected in this graph, their operations must satisfy a strong consistency condition, while the operations invoked by other nodes are allowed to satisfy a weaker condition. The second contribution is the use of such a graph to provide a generic approach to the hybridization of data consistency conditions into the same system. We illustrate this approach on sequential consistency and causal consistency, and present a model in which all data operations are causally consistent, while operations by neighboring processes in the proximity graph are sequentially consistent. The third contribution of the paper is the design and the proof of a distributed algorithm based on this proximity graph, which combines sequential consistency and causal consistency (the resulting condition is called fisheye consistency). In doing so the paper not only extends the domain of consistency conditions, but provides a generic provably correct solution of direct relevance to modern georeplicated systems
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