We consider a family \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} of \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} -periodic Schrdinger operators with \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} -interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} surfaces. We show that in the limit when \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} and the interactions strengths are appropriately scaled, \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} has at most \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} gaps within finite intervals, and moreover, the limiting behavior of the first \{\mathcal {H}^arepsilon\} _ {arepsilon> 0} gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.

We investigate spectral properties of the Laplacian in <i>L</i>² (<i>Q</i>), where <i>Q</i> is a tubular region in ³ of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary situations, when the tube is bent but not twisted, and secondly, it is twisted but not bent. In the first case we derive sufficient conditions for the presence and absence of the discrete spectrum showing, roughly speaking, that they depend on the direction in which the tube is bent. In the second case we show that a constant twist raises the threshold of the essential spectrum and a local slowndown of it gives rise to isolated eigenvalues. Furthermore, we prove that the spectral threshold moves up also under a sufficiently gentle periodic twist.

S-transformation is a unitary transformation accomplishing the main part of the diagonalization of the interaction Hamiltonian in strong coupling theories. For a compact symmetry group the action of the S-transformation on the integrals of motion is derived.

Among approaches to defining the Feynman integral, various sequential methods are probably the most popular ones. Here we consider that one based on product formulae, the original idea of which belongs to Nelson71'. This method is powerful in the sense that it enables to prove the strict version of Feynman's basic dynamical formula for a wide class of poten tials^^/including time-dependent ones/4/in comparison with other known methods (see, eg, refs./5, 6/). 0n the other, hand, the method does not correspond exactly to Feynman's heuristic prescription because it replaces the exact action along poly gonal paths by its Riemannian approximation. Moreover, one can regard as an unsatisfactory feature that it employs the" equidistant" time-interval partitions only. In the present paper we show that the last mentioned sour ce of arbitrariness can be removed: the product formula for perturbations of propagators by

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.