Finiteness of integrable n-dimensional homogeneous polynomial potentials

We consider natural Hamiltonian systems of n>1$n>1$ degrees of freedom with polynomial homogeneous potentials of degree k. We show that under a genericity assumption, for a fixed k, at most only a finite number of such systems is integrable. We also explain how to find explicit forms of these integrable potentials for small k.     

Some properties of the n-dimensional plane symmetric solution

An n-dimensional static plane symmetric solution of Einstein field equation, which is judged as the source of n-dimensional Taub solution, is presented in our previous work. The properties of geodesics of this solution are studied in this Letter. The essence of the source is also investigated. A phantom with dust and photon is suggested as the substance of the source matter.     

On representations of the groups Sp(n, 1) and Sp(n)

New classes of unitary irreducible representations of Sp(n, 1) which can be useful for applications in physics are obtained. The infinitesimal operators of these representations of Sp(n, 1) and of irreducible representations of Sp(n+1) with highest weights (m, m, m₃,…,m_n+1) and (m₁, m₂, 0,…,0) are expressed in terms of the simple Clebsch–Gordancoefficients for Sp(n). For Sp(3) and Sp(2, 1) they are found in an explicit form.     

Linearization of a class of n coupled Bernoulli equations

Analytic linearization maps are found for autonomous systems of n coupled Bernoulli equations of a certain class. The class of equations is determined by the requirement that the nonlinear system linearize to its linear part. These equations have applications as models for interacting biological systems.     

On the universality class of certain string theory hadrons

Exploiting the gauge/gravity correspondence we find the spectrum of hadronic-like bound states of adjoint particles with a large global charge in several confining theories. In particular, we consider an embedding of four-dimensional N=1 supersymmetric Yang–Mills into IIA string theory, two classes of three-dimensional gauge theories and the softly broken version of one of them. In all cases we describe the low energy excitations of a heavy hadron with mass proportional to its global charge. These excitations include: the hadron's nonrelativistic motion, its stringy excitations and excitations corresponding to the addition of massive constituents. Our analysis provides ample evidence for the universality of such hadronic states in confining theories admitting supergravity duals. Besides, we find numerically a new smooth solution that can be thought of as a nonsupersymmetric deformation of G₂ holonomy manifolds.     

On semipositive definiteness of 2n-degree forms

The aim of this paper is to present necessary and sufficient conditions for the semipositive definiteness of 2n-degree forms. These conditions allow to verify whether a given map Λ: $\text{A}$ → $\text{A}$ (where $\text{A}$ is the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$) is a semipositive map.     

On the Fefferman class of metrics associated with a three-dimensional CR space

We derive the explicit forms of Fefferman's metric for a Cauchy-Riemann space admitting a solution of the tangential Cauchy-Riemann equation and of the corresponding Weyl tensor. We show that its Petrov type is 0 in the case of the hyperquadric or N in all other cases, and that the Fefferman class of metrics does not contain any nonflat solution of Einstein's vacuum equations with cosmological constant.Work supported in part by the Polish Ministry of Science and Higher Education, Research Problem CPBP 01.03.     

On a resolution of the multiplicity problem for U(n)

The Borel-Weil theory of holomorphic induction is used to provide a resolution of the multiplicity problem occuring in the tensor product decomposition of representations of the U(n) groups.     

On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order

We review recently developed new powerful techniques to compute a class of Feynman diagrams at any loop order, known as sunrise-type diagrams. These sunrise-type topologies have many important applications in many different fields of physics and we believe it to be timely to discuss their evaluation from a unified point of view. The method is based on the analysis of the diagrams directly in configuration space which, in the case of the sunrise-type diagrams and diagrams related to them, leads to enormous simplifications as compared to the traditional evaluation of loops in momentum space. We present explicit formulae for their analytical evaluation for arbitrary mass configurations and arbitrary dimensions at any loop order. We discuss several limiting cases in their kinematical regimes which are e.g. relevant for applications in HQET and NRQCD. We completely solve the problem of renormalization using simple formulae for the counterterms within dimensional regularization. An important application is the computation of the multi-particle phase space in D-dimensional space-time which we discuss. We present some examples of their numerical evaluation in the general case of D-dimensional space-time as well as in integer dimensions D = D₀ for different values of dimensions including the most important practical cases D₀ = 2, 3, 4. Substantial simplifications occur for odd integer space-time dimensions where the final results can be expressed in closed form through elementary functions. We discuss the use of recurrence relations naturally emerging in configuration space for the calculation of special series of integrals of the sunrise topology. We finally report on results for the computation of an extension of the basic sunrise topology, namely the spectacle topology and the topology where an irreducible loop is added.     

Integrable perturbations of the N-dimensional isotropic oscillator

Two new families of completely integrable perturbations of the N-dimensional isotropic harmonic oscillator Hamiltonian are presented. Such perturbations depend on arbitrary functions and N free parameters and their integrals of motion are explicitly constructed by making use of an underlying h₆-coalgebra symmetry. Several known integrable Hamiltonians in low dimensions are obtained as particular specializations of the general results here presented. An alternative route for the integrability of all these systems is provided by a suitable canonical transformation which, in turn, opens the possibility of adding (N−1) ‘Rosochatius’ terms that preserve the complete integrability of all these models.     

Solid-state fluorescence property and crystal structure of biphenyl derivatives with carboxyl and n-alkyl groups

By introducing carboxyl and n-alkyl groups into a biphenyl molecule, we can increase the photoluminescence quantum yields of biphenyl derivatives in the solid state. The solid-state fluorescence maxima (λ_em) of 4-biphenylcarboxylic acid derivatives change according to the n-alkyl group introduced.     

Adiabatic sound velocity and compressibility of a trapped d-dimensional ideal anyon gas

The adiabatic sound velocity and compressibility for harmonically trapped ideal anyons in arbitrary dimensions are calculated within Haldane fractional exclusion statistics. The corresponding low-temperature and high-temperature behaviors are studied in detail. To compare with the experimental result of unitary fermions, the sound velocity for anyons in the cigar-shaped trap is derived. The sound velocity for anyons in the disk-shaped trap is also calculated. With the parameter g=0.287$g=0.287$, the sound velocity of cigar-shaped unitary fermions modeled by anyons is in good agreement with the experimental result, while that of disk-shaped unitary fermions is v₀/vF=0.406$v_0/v_F=0.406$ with Fermi velocity vF$v_F$.     

A few properties of a certain class of degenerate space-times

The properties are studied of a class of space-times determined by assuming the shape of the metric formds ² including disposable coordinate functions. It has been found that this class includes degenerate space-times with geodetic, null, shear-free congruences with nonvanishing expansion. The theorem has been proved that this class of solutions of the Einstein equations can easily be expanded to solutions of Einstein-Maxwell equations with a fairly general electromagnetic field. For a selected subclass relations are given between the functions determining the metric form, and two new explicit solutions with arbitrary functions of the Einstein-Maxwell equations with a cosmological constant are found.On leave from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

On a class of generalized Poincaré groups: InhomogeneousSL(n,C)

A certain class of non semi-simple Lie groupsISL(n, C) based onSL(n, C) is investigated. Its Lie algebra and invariants are determined. The connection betweenISL(2,C) and the Poincaré group is discussed.On leave from Université de Marseille, Institut de Physique Théorique.     

On the finiteness of discrete spectrum in the n-particle problem

In spaces of functions of any given symmetry sufficient conditions have been obtained for the discrete spectrum finiteness of the energy operators of many particle quantum systems in the absence of external fields.The most general case is considered when the beginning of the essential spectrum of the Hamiltonian is defined by breaking the initial system into any number of stable subsystems. Results obtained are applicable, in particular, to systems with a short-range and to systems with long-range action (in the case of interaction compensation).