Abstract: Abstract
We extend a randomization method, introduced by Shiffman–Zelditch and developed by Burq–Lebeau on compact manifolds for the Laplace operator, to the case of
\({\mathbb{R}^d}\)
with the harmonic oscillator. We construct measures, thanks to probability laws which satisfies the concentration of measure property, on the support of which we prove optimal-weighted Sobolev estimates on
\({\mathbb{R}^d}\)
. This construction relies on accurate estimates on the spectral function in a non-compact configuration space. As an application, we show that there exists a basis of Hermite functions with good decay properties in
\({L^{\infty}(\mathbb{R}^{d})}\)
, when d ≥ 2. PubDate: 2015-02-01

Abstract: Abstract
We consider the initial value problem for a three-component system of quadratic derivative nonlinear Schrödinger equations in two space dimensions with the masses satisfying the resonance relation. We present a structural condition on the nonlinearity under which small data global existence holds. It is also shown that the solution is asymptotically free. Our proof is based on the commuting vector field method combined with smoothing effects. PubDate: 2015-02-01

Abstract: Abstract
We consider generalised versions of the spin-boson model at small coupling. We assume the spin (or atom) to sit at the origin
\({0 \in \mathbb{R}^d}\)
and the propagation speed v
p
of free bosons to be constant, i.e. independent of momentum. In particular, the bosons are massless. We prove detailed bounds on the mean number of bosons contained in the ball
\({\{ x \leq v_p t \}}\)
. In particular, we prove that, as
\({t \to \infty}\)
, this number tends to an asymptotic value that can be naturally identified as the mean number of bosons bound to the atom in the ground state. Physically, this means that bosons, that are not bound to the atom, are travelling outwards at a speed that is not lower than v
p
, hence the term ‘minimal velocity estimate’. Additionally, we prove bounds on the number of emitted bosons with low momentum (soft mode bounds). This paper is an extension of our earlier work in De Roeck and Kupiainen (Annales Henri Poincaré 14:253–311, 2013). Together with the results in De Roeck and Kupiainen (Annales Henri Poincaré 14:253–311, 2013), the bounds of the present paper suffice to prove asymptotic completeness, as we describe in De Roeck et al. (Asymptotic completeness in the massless spin-boson model, 2012). PubDate: 2015-02-01

Abstract: Abstract
Our previous constructions of Borchers triples are extended to massless scattering with nontrivial left and right components. A massless Borchers triple is constructed from a set of left–left, right–right and left–right scattering functions. We find a correspondence between massless left–right scattering S-matrices and massive block diagonal S-matrices. We point out a simple class of S-matrices with examples. We study also the restriction of two-dimensional models to the lightray. Several arguments for constructing strictly local two-dimensional nets are presented and possible scenarios are discussed. PubDate: 2015-02-01

Abstract: Abstract
We study a class of random block operators which appear as effective one-particle Hamiltonians for the anisotropic XY quantum spin chain in an exterior magnetic field given by an array of i.i.d. random variables. For arbitrary non-trivial single-site distribution of the magnetic field, we prove dynamical localization of these operators at non-zero energy. PubDate: 2015-02-01

Abstract: Abstract
We show the explicit agreement between the derivation of the Bekenstein–Hawking entropy of a Euclidean BTZ black hole from the point of view of spin foam models and canonical quantization. This is done by considering a graph observable (corresponding to the black hole horizon) in the Turaev–Viro state sum model, and then analytically continuing the resulting partition function to negative values of the cosmological constant. PubDate: 2015-02-01

Abstract: Abstract
Starting from the form factor expansion in finite volume, we derive the multidimensional generalization of the so-called Natte series for the time- and distance-dependent reduced density matrix at zero temperature in the non-linear Schrödinger model. This representation allows one to read-off straightforwardly the long-time/large-distance asymptotic behaviour of this correlator. This method of analysis reduces the complexity of the computation of the asymptotic behaviour of correlation functions in the so-called interacting integrable models, to the one appearing in free-fermion equivalent models. We compute explicitly the first few terms appearing in the asymptotic expansion. Part of these terms stems from excitations lying away from the Fermi boundary, and hence go beyond what can be obtained using the CFT/Luttinger liquid-based predictions. PubDate: 2015-02-01

Abstract: Abstract
In this paper, we describe the weak limits of the measures associated to the eigenfunctions of the Laplacian on a Quantum graph for a generic metric in terms of the Gauss map of the determinant manifold. We describe also all the limits with minimal support (the “scars”). PubDate: 2015-02-01

Abstract: Abstract
We construct a family of transforms labeled by (ν, m) and mapping isometrically square integrable functions on a finite subset of
\({\mathbb{R}}\)
onto L
2-eigenspaces associated with the discrete spectrum of a charged particle evolving in the Riemann sphere under influence of a uniform magnetic field with a strength proportional to
\({2\nu \in \mathbb{Z}_{+}^{\ast}}\)
. These transforms are attached to spherical Landau levels
\({\lambda _{m}^{\nu}=\left( 2m+1\right) \nu +m\left( m+1\right)}\)
with
\({m\in \mathbb{Z}_{+}}\)
and are called discrete Bargmann transforms. PubDate: 2015-02-01

Abstract: Abstract
To any solution of a linear system of differential equations, we associate a matrix kernel, correlators satisfying a set of loop equations, and in the presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight
\({\hbar}\)
per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non-trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the qth reductions of KP—which contain the (p, q) models as a specialization. PubDate: 2015-01-13

Abstract: Abstract
In this paper we extend our previous results concerning Jackson integral solutions of the boundary quantum Knizhnik–Zamolodchikov (qKZ) equations with diagonal K-operators to higher spin representations of quantum affine
\({\mathfrak{sl}_2}\)
. First we give a systematic exposition of known results on R-operators acting in the tensor product of evaluation representations in Verma modules over quantum
\({\mathfrak{sl}_2}\)
. We develop the corresponding fusion of K-operators, which we use to construct diagonal K-operators in these representations. We construct Jackson integral solutions of the associated boundary qKZ equations and explain how in the finite-dimensional case they can be obtained from our previous results by the fusion procedure. PubDate: 2015-01-03

Abstract: Abstract
Open Quantum Random Walks, as developed in Attal et al. (J. Stat. Phys. 147(4):832–852, 2012), are a quantum generalization of Markov chains on finite graphs or on lattices. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in quantum information theory (such as the well-known Hadamard random walk). Typically, in the case of open quantum random walks on lattices, their distribution seems to always converge to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous open quantum random walks on
\({\mathbb{Z}^{\rm d},}\)
we prove such a central limit theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on
\({\mathbb{Z}^{\rm d}}\)
times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a central limit theorem with explicit drift and explicit covariance matrix. The interesting point which appears with our construction and result is that it applies actually to a wider setup: it provides a central limit theorem for the sequence of recordings of the quantum trajectories associated wih any completely positive map. This is what we show and develop as an application of our result. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block diagonal in a common basis. PubDate: 2015-01-01

Abstract: Abstract
We investigate various aspects of invariance under local symmetries in the framework of perturbative algebraic quantum field theory (pAQFT). Our main result is the proof that the quantum Batalin–Vilkovisky operator, on-shell, can be written as the commutator with the interacting BRST charge. Up to now, this was proven only for a certain class of fields in quantum electrodynamics and in Yang–Mills theory. Our result is more general and it holds in a wide class of theories with local symmetries, including general relativity and the bosonic string. We also comment on other issues related to local gauge invariance and, using the language of homological algebra, we compare different approaches to quantization of gauge theories in the pAQFT framework. PubDate: 2015-01-01

Abstract: Abstract
Consider a point scatterer (the Laplacian perturbed by a delta-potential) on the standard three-dimensional flat torus. Together with the eigenfunctions of the Laplacian which vanish at the point, this operator has a set of new, perturbed eigenfunctions. In a recent paper, the author was able to show that all of the perturbed eigenfunctions are uniformly distributed in configuration space. In this paper we prove that almost all of these eigenfunctions are uniformly distributed in phase space, i.e. we prove quantum ergodicity for the subspace of the perturbed eigenfunctions. An analogue result for a point scatterer on the two-dimensional torus was recently proved by Kurlberg and Ueberschär. PubDate: 2015-01-01

Abstract: Abstract
We give a partially alternate proof of reality of the spectrum of the imaginary cubic oscillator in quantum mechanics. PubDate: 2015-01-01

Abstract: Abstract
We give a quantitative refinement and simple proofs of mode stability type statements for the wave equation on Kerr backgrounds in the full sub-extremal range ( a < M). As an application, we are able to quantitatively control the energy flux along the horizon and null infinity and establish integrated local energy decay for solutions to the wave equation in any bounded-frequency regime. PubDate: 2015-01-01

Abstract: Abstract
The purpose of this paper is to rigorously investigate the orbital magnetism of core electrons in three-dimensional crystalline ordered solids and in the zero-temperature regime. To achieve that, we consider a non-interacting Fermi gas subjected to an external periodic potential modeling the crystalline field within the tight-binding approximation (i.e., when the distance between two consecutive ions is large). For a fixed number of particles in the Wigner–Seitz cell and in the zero-temperature limit, we derive an asymptotic expansion for the bulk zero-field orbital susceptibility. We prove that the leading term is the superposition of the Larmor diamagnetic contribution, generated by the quadratic part of the Zeeman Hamiltonian, together with the ‘complete’ orbital Van Vleck paramagnetic contribution, generated by the linear part of the Zeeman Hamiltonian, and related to field-induced electronic transitions. PubDate: 2015-01-01

Abstract: Abstract
We establish reflection positivity for Gibbs trace states defined by certain Hamiltonians that describe the interaction of Majoranas on a lattice. These Hamiltonians may include many-body interactions, as long as the signs of the associated coupling constants satisfy certain restrictions. We show that reflection positivity holds on an even sub-algebra of Majoranas. PubDate: 2015-01-01

Abstract: Abstract
We analyze pull-in instability of electrostatically actuated microelectromechanical systems, and we find that as the device size is reduced, the effect of the Casimir force becomes more important. In the miniaturization process there is a minimum size for the device below which the system spontaneously collapses with zero applied voltage. According to the mathematical analysis, we obtain a set U in the plane, such that elements of U correspond to minimal stable solutions of a two-parameter mathematical model. For points on the boundary
\({\Upsilon}\)
of U, there exists weak solutions to this model, which are called extremal solutions. More refined properties of stable solutions—such as regularity, stability, uniqueness—are also established. PubDate: 2015-01-01