We investigate a periodic quantum graph in form of a square lattice with a
general self-adjoint coupling at the vertices. We analyze the spectrum, in
particular, its high-energy behaviour. Depending on the coupling type, bands
and gaps have different asymptotics. Bands may be flat even if the edges are
coupled, and non-flat band widths may behave as , as the ...
We examine scale invariant Fulop-Tsutsui couplings in a quantum vertex of a
general degree . We demonstrate that essentially same scattering amplitudes
as for the free coupling can be achieved for two -parameter
Fulop-Tsutsui subfamilies if is odd, and for three -parameter
Fulop-Tsutsui subfamilies if is even. We also work up an approximation
scheme for a general Fulop-...
A word defined over an alphabet is -balanced
() if for all pairs of factors , of of the same
length and for all letters , the difference between the number
of letters in and is less or equal to . In this paper we
consider a ternary alphabet and a class of
substitutions defined by $\p...
J. Phys. Soc. Jpn. 78 (2009) 124004 (7 pages) Dec 2009 [Refereed]
We examine scattering properties of singular vertex of degree and , taking advantage of a new form of representing the vertex boundary
condition, which has been devised to approximate a singular vertex with finite
potentials. We show that proper identification of and
components in the connection condition between outgoing lines enables the
designing of quantum...
Annals of Physics 325 (2010) 548-578 Feb 2010 [Refereed]
The longstanding open problem of approximating all singular vertex couplings
in a quantum graph is solved. We present a construction in which the edges are
decoupled; an each pair of their endpoints is joined by an edge carrying a potential and a vector potential coupled to the "loose" edges by a coupling. It is shown that if the lengths of the connecting edges
shrin...
J. Phys. A: Math. Theor. 41 415206- Jul 2008 [Refereed]
We study Schr\"odinger operators on an infinite quantum graph of a chain form
which consists of identical rings connected at the touching points by -couplings with a parameter . If the graph is "straight",
i.e. periodic with respect to ring shifts, its Hamiltonian has a band spectrum
with all the gaps open whenever . We consider a "bending&q...
We discuss approximations of the vertex coupling on a star-shaped quantum
graph of edges in the singular case when the wave functions are not
continuous at the vertex and no edge-permutation symmetry is present. It is
shown that the Cheon-Shigehara technique using interactions with
nonlinearly scaled couplings yields a -parameter family of boundary
conditions in the sense of no...
Theoret. Informatics Appl. 41 307-328 Aug 2007 [Refereed]
We study arithmetical and combinatorial properties of -integers for being the root of the equation . We determine with the accuracy of the maximal number of -fractional positions, which may arise as a result of addition of two -integers. For the infinite word coding distances between
consecutive $\beta...
Proc. NSF Research Conference “Quantum Graphs and Their Applications”, Snowbird (2005); AMS “Contemporary Mathematics” Series, Providence 415 109-120 Apr 2006
We consider boundary conditions at the vertex of a star graph which make
Schroedinger operators on the graph self-adjoint, in particular, the
two-parameter family of such conditions invariant with respect to permutations
of graph edges. It is proved that the corresponding operators can be
approximated in the norm-resolvent sense by elements of another Schroedinger
operator family on the same gr...
