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.

We analyze a family of singular Schrdinger operators with local singular interactions supported by a hypersurface n, n 2, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of the interaction is characterized by four real parameters, the earlier studied case of - and -interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.

The paper is devoted to a model of a mesoscopic system consisting of a pair of parallel planar waveguides separated by an infinitely thin semitransparent boundary modeled by a transverse interaction. We develop the BirmanSchwinger theory for the corresponding generalized Schrdinger operator. The spectral properties become nontrivial if the barrier coupling is not invariant with respect to longitudinal translations, in particular, there are bound states if the barrier is locally more transparent in the mean and the coupling parameter reaches the same asymptotic value in both directions along the guide axis. We derive the weak-coupling expansion of the ground-state eigenvalue for the cases when the perturbation is small in the supremum and the <b>L</b>¹-norms. The last named result applies to the situation when the support of the leaky part shrinks: the obtained asymptotics differs from that of a double guide divided