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ALMOST PERIODIC SCHRÖDINGER OPERATORS II. THE INTEGRATED DENSITY OF STATES
 VOL. 50, NO. DUKE MATHEMATICAL JOURNAL (C) MARCH 1983
, 1983
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qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 73 (3 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation 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 qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian 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 qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Analysis of an importance sampling estimator for tandem queues
 ACM Transactions on Modeling and Computer Simulation
, 1995
"... We analyze the performance of an importance sampling estimator for a rareevent 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 t ..."
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Cited by 53 (1 self)
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We analyze the performance of an importance sampling estimator for a rareevent 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 rareevent simulation problem in which boundaries on the state space play a significant role.
A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure
 Ann. of Math
, 2000
"... Abstract. We continue the study of the Aamplitude associated to a halfline Schrödinger operator, − d2 dx2 + q in L2 ((0, b)), b ≤ ∞. A is related to the WeylTitchmarsh mfunction 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 t ..."
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Cited by 50 (20 self)
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Abstract. We continue the study of the Aamplitude associated to a halfline Schrödinger operator, − d2 dx2 + q in L2 ((0, b)), b ≤ ∞. A is related to the WeylTitchmarsh mfunction 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 Aamplitude 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 mfunctions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal WeylTitchmarsh mfunction. Finally, we discuss some examples where one can compute A exactly. 1.
NonEquilibrium Steady States of Finite Quantum Systems Coupled to Thermal Reservoirs
 COMMUN. MATH. PHYS
, 2001
"... We study the nonequilibrium 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 ..."
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Cited by 50 (8 self)
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We study the nonequilibrium 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 nonequilibrium 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].
R.: Derivation of the GrossPitaevskii Equation for rotating Bose gases
 Comm. Math. Phys
, 2006
"... We prove that the GrossPitaevskii equation correctly describes the ground state energy and corresponding oneparticle density matrix of rotating, dilute, trapped Bose gases with repulsive twobody interactions. We also show that there is 100 % BoseEinstein condensation. While a proof that the GP e ..."
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Cited by 35 (4 self)
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We prove that the GrossPitaevskii equation correctly describes the ground state energy and corresponding oneparticle density matrix of rotating, dilute, trapped Bose gases with repulsive twobody interactions. We also show that there is 100 % BoseEinstein condensation. While a proof that the GP equation correctly describes nonrotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state. 1
Risk communication
 Proceedings of the national conference on risk communication, Conservation Foundation,Washington, DC
, 1987
"... We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, O<c<[(l/2)(n2)] * with H>O. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a fo ..."
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Cited by 34 (1 self)
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We consider Schrodinger semigroups e. IH, H =A+V on Iw ” with VcIxl ’ as 1x1rco, O<c<[(l/2)(n2)] * with H>O. We determine the exact power law divergence of I~e‘Hi~p,p and of some IIe‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x) 1.x‘.:Ta 1991 Academic Press, Inc. 1.
Universality in BlowUp for Nonlinear Heat Equations
"... We consider the classical problem of the blowingup of solutions of the nonlinear heat equation. We show that there exist infinitely many profiles around the blowup point, and for each integer k, we construct a set of codimension 2k in the space of initial data giving rise to solutions that blowup ..."
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Cited by 26 (2 self)
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We consider the classical problem of the blowingup of solutions of the nonlinear heat equation. We show that there exist infinitely many profiles around the blowup point, and for each integer k, we construct a set of codimension 2k in the space of initial data giving rise to solutions that blowup according to the given profile. 1 Introduction We consider the problem of the blowup of solutions of the initial value problem u t = u xx + u p (1) where p ? 1; u = u(x; t); x 2 R, and u(\Delta; 0) = u 0 2 C 0 (R). It is wellknown that, for a large class of initial data u 0 , the solution will diverge in a finite time at a single point (for reviews on this problem, see [9, 13]). We are interested in the profile of the solution at the time of blowup. To explain what this means, let us fix the blowup point to be 0 and the blowup time to be T . Then, we ask whether it is possible to find a function f (x) and a rescaling g(t; T ) so that lim t"T (T \Gamma t) 1 p\Gamma1 u(g(t; ...