32 research outputs found
Irreducibility and reducibility for the energy representation of the group of mappings of a Riemannian manifold into a compact semisimple Lie group
AbstractThe irreducibility of the energy representation of the group of smooth mappings from a Riemannian manifold of dimension d ⩾ 3 into a compact semisimple Lie group is proven. The nonequivalence of the representations associated with different Riemann structures is also proven for d ⩾ 3. The case d = 2 is examined and irreducibility and nonequivalence results are also given. The reducibility in the case d = 1 is pointed out (in this case the commutant contains a representation equivalent with the energy representation)
Factoriality of representations of the group of paths on SU(n)
AbstractFactoriality in the cyclic component of the vacuum for the energy representation of SU(n)-valued paths groups is proved. The main tool is a lemma concerning generic pairs of Cartan subalgebras in the Lie algebra su(n) of SU(n) groups
Quantum electrodynamics of relativistic bound states with cutoffs
We consider an Hamiltonian with ultraviolet and infrared cutoffs, describing
the interaction of relativistic electrons and positrons in the Coulomb
potential with photons in Coulomb gauge. The interaction includes both
interaction of the current density with transversal photons and the Coulomb
interaction of charge density with itself. We prove that the Hamiltonian is
self-adjoint and has a ground state for sufficiently small coupling constants.Comment: To appear in "Journal of Hyperbolic Differential Equation
Asymptotic behaviour of the spectrum of a waveguide with distant perturbations
We consider the waveguide modelled by a -dimensional infinite tube. The
operator we study is the Dirichlet Laplacian perturbed by two distant
perturbations. The perturbations are described by arbitrary abstract operators
''localized'' in a certain sense, and the distance between their ''supports''
tends to infinity. We study the asymptotic behaviour of the discrete spectrum
of such system. The main results are a convergence theorem and the asymptotics
expansions for the eigenvalues. The asymptotic behaviour of the associated
eigenfunctions is described as well. We also provide some particular examples
of the distant perturbations. The examples are the potential, second order
differential operator, magnetic Schroedinger operator, curved and deformed
waveguide, delta interaction, and integral operator
Euclidean Gibbs states of interacting quantum anharmonic oscillators
A rigorous description of the equilibrium thermodynamic properties of an
infinite system of interacting -dimensional quantum anharmonic oscillators
is given. The oscillators are indexed by the elements of a countable set
, possibly irregular; the anharmonic potentials
vary from site to site. The description is based on the representation of the
Gibbs states in terms of path measures -- the so called Euclidean Gibbs
measures. It is proven that: (a) the set of such measures
is non-void and compact; (b) every obeys an
exponential integrability estimate, the same for the whole set
; (c) every has a
Lebowitz-Presutti type support; (d) is a singleton at
high temperatures. In the case of attractive interaction and we prove
that at low temperatures. The uniqueness of Gibbs
measures due to quantum effects and at a nonzero external field are also proven
in this case. Thereby, a qualitative theory of phase transitions and quantum
effects, which interprets most important experimental data known for the
corresponding physical objects, is developed. The mathematical result of the
paper is a complete description of the set , which refines
and extends the results known for models of this type.Comment: 60 page