We discuss formulations of boundary conditions in a quantum graph vertex and demonstrate that the so-called ST-form can be further reduced up to a form more effective in certain applications: In particular, in identifying the number of independent parameters for given ranks of two connection matrices, or in calculating the scattering matrix when both matrices are singular. The new form of boundary conditions, called the P Q R S-form, also gives a natural scheme to design generalized low and high pass quantum filters.

In this paper, we analyze a free quantum particle in a straight Dirichlet waveguide which has at its axis two Dirichlet barriers of lengths separated by a window of length 2a. It is known that if the barriers are semi-infinite, i.e., we have two adjacent waveguides coupled laterally through the boundary window, the system has for any a > 0 a finite number of eigenvalues below the essential spectrum threshold. Here, we demonstrate that for large but finite the system has resonances which converge to the said eigenvalues as , and derive the leading term in the corresponding asymptotic expansion.

The concept of matrix canonical realization of a Lie algebra is introduced. The generators of the Lie algebra of the pseudoorthogonal group n) are recurrently expressed in terms of matrices with polyno-mial elements in a certain number of quantum-mechanical canonical variables p~, qi and they depend on a certain number of free real para-meters. The realizations are, in the well-defined sense, skew-hermitean and Casimir operators are multiples of the identity element. Part of them are usual canonical realizations.

The ideal anti-Zeno effect means that a perpetual observation leads to an immediate disappearance of the unstable system. We present a straightforward way to derive sufficient conditions under which such a situation occurs expressed in terms of the decaying states and spectral properties of the Hamiltonian. They show, in particular, that the gap between Zeno and anti-Zeno effects is in fact very narrow.

We consider a model of leaky quantum wires in three dimensions. The Hamiltonian is a singular perturbation of the Laplacian supported by a line with the coupling which is bounded and periodically modulated along the line. We demonstrate that such a system has a purely absolutely continuous spectrum and its negative part has band structure with an at most finite number of gaps. This result is extended also to the situation when there is an infinite number of the lines supporting the perturbations arranged periodically in one direction.