191,279 research outputs found
Spectral characterizations of propeller graphs
A propeller graph is obtained from an -graph by attaching a path to
the vertex of degree four, where an -graph consists of two cycles with
precisely one common vertex. In this paper, we prove that all propeller graphs
are determined by their Laplacian spectra as well as their signless Laplacian
spectra
Band spectra of rectangular graph superlattices
We consider rectangular graph superlattices of sides l1, l2 with the
wavefunction coupling at the junctions either of the delta type, when they are
continuous and the sum of their derivatives is proportional to the common value
at the junction with a coupling constant alpha, or the "delta-prime-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 theta := l1/l2. If the latter is an irrational badly
approximable by rationals, delta lattices have no gaps in the weak-coupling
case. We show that there is a quantization for the asymptotic critical values
of alpha at which new gap series open, and explain it in terms of
number-theoretic properties of theta. We also show how the irregularity is
manifested in terms of Fermi-surface dependence on energy, and possible
localization properties under influence of an external electric field.
KEYWORDS: Schroedinger operators, graphs, band spectra, fractals,
quasiperiodic systems, number-theoretic properties, contact interactions, delta
coupling, delta-prime coupling.Comment: 16 pages, LaTe
Statistical Properties of Quantum Graph Spectra
A general analytical approach to the statistical description of quantum graph
spectra based on the exact periodic orbit expansions of quantum levels is
discussed. The exact and approximate expressions obtained in \cite{Anima} for
the probability distribution functions using the spectral hierarchy method are
analyzed. In addition, the mechanism of appearance of the universal statistical
properties of spectral fluctuations of quantum-chaotic systems is considered in
terms of the semiclassical theory of periodic orbits.Comment: 14 pages, 3 fi
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