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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.

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.

Properties of the subset of polygonal paths in the Hilbert space of paths referring to a<i>d</i>-dimensional quantum-mechanical system are examined. The results are used to discuss various types of polygonal-path approximations appearing in the functional-integration theory. The uniform approximation is applied to extend the definition of the Feynman maps from our previous paper and to prove consistency of this extension. Relations of the extended<i>F</i> _<i>i</i> map to the Wiener integral are given.

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