A product formula is proved which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P. It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P.

We discuss a model of a leaky quantum wire and a family of quantum dots described by Laplacian in L²(²) with an attractive singular perturbation supported by a line and a finite number of points. The discrete spectrum is shown to be nonempty, and furthermore, the resonance problem can be explicitly solved in this setting; by Birman-Schwinger method it is reformulated into a Friedrichs-type model.

The notion of a vector space is obtained by axiomatization of the properties of the threedimensional space of Euclidean geometry, or of configuration spaces of classical mechanics. A vector (or linear) space V is a set {x, y,...} equipped with the operations of summation,[x, y] x+ y V, and multiplication by a complex or real number ,[, x] x V, such that (i) The summation is commutative, x+ y= y+ x, and associative,(x+ y)+ z= x+(y+ z). There exist a zero element 0 V, and an inverse element x V, to any x V so that x+ 0= x and x+( x)= 0 holds for all x V.(ii) (x)=() x and 1x= x.(iii) The summation and multiplication are distributive, (x+ y)= x+ y and (+ ) x= x+ x.

We investigate non-equilibrium particle transport in a system consisting of a geometric scatterer and two leads coupled to heat baths with different chemical potentials. We derive an expression for the corresponding current, the carriers of which are fermions, and analyze numerically its dependence on the model parameters in examples where the scatterer has a rectangular or triangular shape.

We analyze a family of singular Schrdinger operators describing a Neumann waveguide with a periodic array of singular traps of a delta\prime type. We show that in the limit when perpendicular size of the guide tends to zero and the delta\prime interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.