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Abstract. We examine, for −1 < q < 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) Xt = at + a ∗ t – where the at fulfill the q-commutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

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Abstract. We continue the study of the A-amplitude associated to a half-line Schrödinger operator, − d2 dx2 + q in L2 ((0, b)), b ≤ ∞. A is related to the Weyl-Titchmarsh m-function via m(−κ2) = −κ − ∫ a 0 A(α)e−2ακ dα+O(e −(2a−ε)κ) for all ε> 0. We discuss five issues here. First, we extend the theory to general q in L1 ((0, a)) for all a, including q’s which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure ρ: A(α) = −2 ∫ ∞ 1 λ − 2 sin(2α − ∞ √ λ)dρ(λ) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b < ∞. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly. 1.

We analyze the performance of an importance sampling estimator for a rare-event probability in tandem Jackson networks. The rare event we consider corresponds to the network population reaching K before returning to O, starting from O, with K large. The estimator we study is based on interchanging the arrival rate and the smallest service rate and 1s therefore a generalization of the asymptotically optimal estimator for an M/M/1 queue. We examine its asymptotic performance for large K, showing that in certain parameter regions the estimator has an asymptotic efficiency property, but that in other regions it does not. The setting we consider is perhaps the simplest case of a rare-event simulation problem in which boundaries on the state space play a significant role.

We study the non-equilibrium statistical mechanics of a #-level quantum system, # , coupled to two independent free Fermi reservoirs # # , # # , which are in thermal equilibrium at inverse temperatures # # ## # # .Weprove that, at small coupling, the combined quantum system ### # ## # has a unique non-equilibrium steady state (NESS) and that the approach to this NESS is exponentially fast. We show that the entropy production of the coupled system is strictly positive and relate this entropy production to the heat uxes through the system. A part of our argument is general and deals with spectral theory of NESS. In the abstract setting of algebraic quantum statistical mechanics we introduce the new concept of #- Liouvillean, #, and relate the NESS to zero resonance eigenfunctions of # # . In the specific model ### # ## # we study the resonances of # # using the complex deformation technique developed previously by the authors in [JP1].

this paper is a proxy for what deserves a book or at least a very long review article. In attempting to overview such a vast area in a few pages, I have had to focus on the high points. No proofs are given and I have settled for usually quoting the initial or especially significant papers. I have no doubt that I have left out some important papers, and if so, I ask the forgiveness of the reader (and their authors!).

Payne-Pólya-Weinberger conjecture, Sperner’s inequality, biharmonic operator, bi-Laplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the Pólya-Szegő conjecture, universal inequalities for eigenvalues, Hile-Protter inequality, H. C. Yang’s inequality. Short title: Isoperimetric and Universal Inequalities This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet ” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to

We extend the trace formula recently proven for general one-dimensional Schrödinger operators which obtains the potential V (x) from a function ξ(x, λ) by deriving trace relations computing moments of ξ(x, λ) dλ in terms of polynomials in the derivatives of V at x. We describe the relation of those polynomials to KdV invariants. We also discuss trace formulae for analogs of ξ associated with boundary conditions other than the Dirichlet boundary condition underlying ξ.

We consider a number of simple quantum Hamiltonians H(-iV, x) with the following property: H(-rV, x) has discrete spectrum even though {(p, q) 1 H(p, q) < E} has infinite volume. 1.