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We consider a charged spinless quantum particle confined to a graph consisting of a loop to which a halfline lead is attached; this system is placed into a homogeneous magnetic field perpendicular to the loop plane. We derive the reflection amplitude and show that there is an infinite ladder of resonances; analyzing the resonance pole trajectories, we show that half of them turn into true embedded eigenvalues provided the flux through the loop is an integer or half-integer multiple of the flux unit hc/e. We also describe a general method to solve the scattering problem on graphs of which the present model is a simple particular case. Finally, we discuss ways in which a state localized initially at the loop decays.

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The Feynman maps introduced first by Truman are examined. The domain considered here consists of the Fresnel-integrable functions in the sense of Albeverio and Hoegh-Krohn; extensions to wider classes of functions will be studied in the following paper. The original definition of the F-maps is slightly modified: we start from the underlying measures on the Hilbert space of paths in order to avoid use of improper integrals. Some new properties of the F-maps are derived. In particular, the dominated convergence theorem is shown to be not valid for the F₁-map (or Feynman integral); this fact is of a certain importance for the classical limit of quantum mechanics.

Some electromagnetic materials exhibit, in a given frequency range, effective dielectric permittivity and/or magnetic permeability which are negative. In the literature, they are called negative index materials, left-handed materials or meta-materials. We propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta-material. The method we investigate can be considered as an alternative method compared to the method presented by the second author and co-workers. In particular, we shall use the abstract framework they developed to prove well-posedness of the exact problem. We recast this problem to fit later discretization by the staggered discontinuous Galerkin method developed by the first author and co-worker, a method which relies on introducing an auxiliary unknown. Convergence of the numerical method is proven, with the help of explicit infsup