We study a free quantum motion on periodically structured manifolds composed of elementary two-dimensional'cells' connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way, or two-dimensional square lattice'carpets'. We show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies.

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 two-dimensional particle of charge e interacting with a homogeneous magnetic field perpendicular to the plane and a potential well which is transported along a closed loop in the plane. We show that a bound state corresponding to a simple isolated eigenvalue acquires at that Berry's phase equal to 2 sgn e times the number of flux quanta through the oriented area encircled by the loop. We also argue that this is a purely quantum effect since the corresponding Hannay angle is zero.

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We consider rectangular graph superlattices of sides l 1, l 2 with the wave-function coupling at the junctions either of the type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant , or the s type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio := l 1/l 2. If the latter is an irrational badly approximable by rationals, lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of at which new gap series open, and explain it in terms of number-theoretic properties of . We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible