Local Resolvent Estimates for <Emphasis Type="Italic">N</Emphasis>-body Stark Hamiltonians

For an N-body Stark Hamiltonian , the resolvent estimate holds uniformly in with Re and Im , where , and is a compact interval. This estimate is well known as the limiting absorption principle. In this paper, we report that by introducing the localization in the configuration space, a refined resolvent estimate holds uniformly in with Re and Im . Dedicated to Professor Hideo Tamura on the occasion of his 60th birthday     

Quantization of Semi-Classical Twists and Noncommutative Geometry

A problem of defining the quantum analogues for semi-classical twists in U()[[t]] is considered. First, we study specialization at q = 1 of singular coboundary twists defined in Uq ())[[t]] for g being a nonexceptional Lie algebra, then we consider specialization of noncoboundary twists when = and obtain q-deformation of the semiclassical twist introduced by Connes and Moscovici in noncommutative geometry. Mathematics Subject Classification: 16W30, 17B37, 81R50     

Dynamical Noncommutativity and Noether Theorem in Twisted $${\phi^{\star 4}}$$ Theory

A -product is defined via a set of commuting vector fields , and used in a theory coupled to the fields. The -product is dynamical, and the vacuum solution , reproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy–momentum and angular momentum tensors are explicitly derived.      

Compound Basis Arising from the Basic {A^{(1)}_{1}}-Module

Given a conditionally completely positive map on a unital *-algebra , we find an interesting connection between the second Hochschild cohomology of with coefficients in the bimodule of adjointable maps, where M is the GNS bimodule of , and the possibility of constructing a quantum random walk [in the sense of (Attal et al. in Ann Henri Poincar 7(1):59–104, 2006; Lindsay and Parthasarathy in Sankhya Ser A 50(2):151–170, 1988; Sahu in Quantum stochastic Dilation of a class of Quantum dynamical Semigroups and Quantum random walks. Indian Statistical Institute, 2005; Sinha in Banach Center Publ 73:377–390, 2006)] corresponding to . D. Goswami was supported by a project funded by the Indian National Academy of Sciences. L. Sahu had research support from the National Board of Higher Mathematics, DAE (India) is gratefully acknowledged.     

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral <Emphasis Type="Italic">su</Emphasis>(2) WZNW Model

A zero modes’ Fock space is constructed for the extended chiral WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra at an even root of unity, and of its infinite dimensional extension by the Lusztig operators We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of A central result is the characterization of the Grothendieck ring of both and in Theorem 3.1. The properties of the fusion ring in are related to the braiding properties of correlation functions of primary fields of the conformal current algebra model.      

Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation

Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e^−itH D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation . This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

Can One See the Fundamental Frequency of a Drum?

We establish two-sided estimates for the fundamental frequency (the lowest eigenvalue) of the Laplacian in an open set with the Dirichlet boundary condition. This is done in terms of the interior capacitary radius of Ω which is defined as the maximal possible radius of a ball B with a negligible intersection with the complement of Ω. Here negligibility of means that cap(F)≤ γ cap (B), where cap a means the Wiener (harmonic) capacity and is arbitrarily fixed with the sole restriction . We provide explicit values of constants in the two-sided estimates.     

Cohomology of $${\mathfrak{osp}}(1|2)$$ Acting on Linear Differential Operators on the Supercircle S1|1

We compute the first cohomology spaces of the Lie superalgebra with coefficients in the superspace of linear differential operators acting on weighted densities on the supercircle S ^1|1. The structure of these spaces was conjectured in (Gargoubi et al. in Lett Math Phys 79:5165, 2007). In fact, we prove here that the situation is a little bit more complicated.      

Double Product Integrals and Enriquez Quantisation of Lie Bialgebras II: The Quantum Yang–Baxter Equation

For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra , let be the vector space of tensors over equipped with the Itô Hopf algebra structure derived from the associative multiplication in . It is shown that a necessary and sufficient condition that the double product integral satisfy the quantum Yang–Baxter equation over is that satisfy the same equation over the unital associative algebra got by adjoining a unit element to . In particular, the first-order coefficient r₁ of r[h] satisfies the classical Yang–Baxter equation. Using the fact that the multiplicative inverse of is where is the inverse of in we construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on in the unital associative subalgebra of consisting of formal power series whose zero order coefficient lies in the space of symmetric tensors. The deformation coproduct acts on by conjugating the undeformed coproduct by and the coboundary structure r of is given by where is the flip.Mathematical Subject Classification (2000). 53D55, 17B62     

Polynômes 3-Tangentiels Et Solutions t-Périodiques de KdV

Nous étudions, quel que soit le réseau , les courbes hyperelliptiques donnant lieu, via le dictionnaire de Krichever et la formule d’Its-Mateev, à des solutions méromorphes Λ-doublement périodiques en t de l’équation de Korteweg-de Vries. Ce sont des revêtements marqués finis particuliers de la courbe elliptique (X,q)=(C /Λ,0) que nous nommons paires osculatrices hyperelliptiques. Nous sommes amenés à définir la classe des polynômes 3-tangentiels symétriques et à considérer une surface algébrique réglée S→ X et la surface obtenue par un éclatement en huit points de S. Nous associons alors aux polynômes 3-tangentiels symétriques des diviseurs sur S et . En étudiant ces diviseurs, nous démontrons que les paires osculatrices non-ramifiées au point marqué se factorisent via et reconstruisons ensuite de telles paires sur sous certaines conditions numériques.     

Spectrum of Time Operators

Let H be a self-adjoint operator on a complex Hilbert space . A symmetric operator T on is called a time operator of H if, for all , (D(T) denotes the domain of T) and . In this paper, spectral properties of T are investigated. The following results are obtained: (i) If H is bounded below, then σ(T), the spectrum of T, is either (the set of complex numbers) or . (ii) If H is bounded above, then is either or . (iii) If H is bounded, then . The spectrum of time operators of free Hamiltonians for both nonrelativistic and relativistic particles is exactly identified. Moreover spectral analysis is made on a generalized time operator. This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from the JSPS.     

A Space–Time Integral Estimate For A Large Data Semi-linear Wave Equation on the Schwarzschild Manifold

We consider the wave equation with in . The wave equation on a spherically symmetric manifold with a single closed geodesic surface or on the exterior of the Schwarzschild manifold can be reduced to this form. Using a smoothed Morawetz estimate which does not require a spherical harmonic decomposition, we show that there is decay in for initial data in the energy class, even if the initial data is large. This requires certain conditions on the potentials V, V _L and f. We show that a key condition on the weight in the smoothed Morawetz estimate can be reduced to an ODE condition, which is verified numerically.      

Integrable Discrete Nets in Grassmannians

We consider discrete nets in Grassmannians , which generalize Q-nets (maps with planar elementary quadrilaterals) and Darboux nets (-valued maps defined on the edges of such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.      

Quadratic Deformations of Lie–Poisson Structures

In this letter, first we give a decomposition for any Lie–Poisson structure associated to the modular vector. In particular, splits into two compatible Lie–Poisson structures if . As an application, we classified quadratic deformations of Lie– Poisson structures on up to linear diffeomorphisms. Research partially supported by NSF of China and the Research Project of “Nonlinear Science”.     

Non-relativistic Limit of a Dirac Polaron in Relativistic Quantum Electrodynamics

A quantum system of a Dirac particle interacting with the quantum radiation field is considered in the case where no external potentials exist. Then the total momentum of the system is conserved and the total Hamiltonian is unitarily equivalent to the direct integral of a family of self-adjoint operators acting in the Hilbert space , where is the Hilbert space of the quantum radiation field. The fiber operator is called the Hamiltonian of the Dirac polaron with total momentum . The main result of this paper is concerned with the non-relativistic (scaling) limit of . It is proven that the non-relativistic limit of yields a self-adjoint extension of a Hamiltonian of a polaron with spin 1/2 in non-relativistic quantum electrodynamics.     

Affine Crystal Graphs And Two-Colored Partitions

We define the cores and the quotients of the elements in the crystal graph of the basic representation of or , and then obtain new combinatorial interpretations of the character and the string function formula. Mathematics Subject Classifications (2000). 17B37, 05A17.     

Ion Mobility Measurements and Ion Chemical Reaction Studies at Heavy Elements in a Buffer Gas Cell

Drift time measurements of ions in a buffer gas cell filled with argon have been performed from which changes of the ion mobility and ionic radii for various heavy elements and their compounds were determined. The ionic radius of americium shrinks by (3.1 1.3)% with respect to that of plutonium, and an increase of the radius by (28 2)% of plutonium oxide with respect to plutonium was found. Ion chemical reactions of erbium ions were studied online in an argon buffer gas cell to which the reaction gases oxygen (O) and methane (CH) were added. The erbium ions were implanted into the buffer gas cell with an energy of 50 MeV. The online measured reaction constant = (3.2 0.4) 10 cm/(molecule s) for the reaction Er + O ErO + O agrees with a reference measurement = (3.6 0.4) 10 cm/(molecule s), performed with a Fourier-Transform-Mass-Spectrometer.