J. Phys. A: Math. Theor. 51(28) 285301 Jul 2018 [Refereed]

The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the δ type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the interpolation family ...

J. Phys. A: Math. Theor. 50(45) 455201 Nov 2017 [Refereed]

The paper is concerned with the number of open gaps in spectra of periodic quantum graphs. The well-known conjecture by Bethe and Sommerfeld (1933) says that the number of open spectral gaps for a system periodic in more than one direction is fini...

In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamil...

J. Phys. A: Math. Theor. 50(24) 245304 Jun 2017 [Refereed]

We demonstrate that a complex equiangular tight frame composed of N vectors in dimension d, denoted ETF (d, N), exists if and only if a certain bistochastic matrix, univocally determined by N and d, belongs to a special class of unistochastic matr...

Functional Analysis and Operator Theory for Quantum Physics (EMS Proceedings, Pavel Exner Anniversary Volume; eds.: J. Dittrich, H. Kovarik, A. Laptev) 543-563 May 2017 [Refereed][Invited]

We examine transmission through a quantum graph vertex to which auxiliary edges with constant potentials are attached. We find a characterization of vertex couplings for which the transmission probability from a given ``input'' line to a given ``o...

We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a δ type coupling in the vertices. Such a quantum graph has nonempty point spectrum with compactly supported eigenfunctions provided...

J. Integer Seq. 18(3) Article 15.3.4-(29 pp.) Feb 2015 [Refereed]

According to a result of Richomme, Saari and Zamboni, the abelian complexity of the Tribonacci word satisfies for each . In this paper we derive an automaton that evaluates the function $\rho...

Abelian complexity of a word u is a function that counts the number of pairwise non-abelian-equivalent factors of u of length n. We prove that for any c-balanced Parry word u, the values of the abelian complexity function can be computed by a fini...

The -bonacci word is a generalization of the Fibonacci word to the -letter alphabet . It is the unique fixed point of the Pisot--type substitution $\varphi_m: 0\to 01,\ 1\to 02,\ \ldots,\ (m-2)\to0(m-1),\text...

Acta Polytechnica 53(5) 410-415 Nov 2013 [Refereed]

We study a family of closed quantum graphs described by one singular vertex of order n=4. By suitable choice of the parameters specifying the singular vertex, we can construct a closed path in the parameter space that physically corresponds to th...

J. Math. Phys. 54(3) 032104-(17 pp.) Mar 2013 [Refereed]

We study the transmission of a quantum particle along a straight input–output line to which a graph Γ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen scale-invariant coupling with a ...

We study the scattering in a quantum star graph with a Fülöp--Tsutsui coupling in its vertex and with external potentials on the lines. We find certain special couplings for which the probability of the transmission between two given lines of ...

We propose a technique for exploring the abelian complexity of recurrent
infinite words, focusing particularly on infinite words associated with Parry
numbers. Using that technique, we give the affirmative answer to the open
question posed by Rich...

EPL - Europhys. Lett. 98 50005 (5 pp.) Jun 2012 [Refereed]

We design two simple quantum devices applicable as an adjustable quantum spectral filter and as a flux controller. Their function is based upon the threshold resonance in a Fülöp-Tsutsui type star graph with an external potential added on one ...

Phys. Lett. A 375(43) 3775-3780 Oct 2011 [Refereed]

Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix . Observing that if has at most two eigenvalues, then the scattering matrix of the vertex is a linear combination of the identit...

We demonstrate how the inverse scattering problem of a quantum star graph can
be solved by means of diagonalization of Hermitian unitary matrix when the
vertex coupling is of the scale invariant (or Fül\H{o}p-Tsutsui) form. This
enables the cons...

We discuss formulations of boundary conditions in a quantum graph vertex and demonstrate that the so-called -form can be further reduced up to a form more effective in certain applications: In particular, in identifying the number of independe...

J. Phys. A: Math. Theor. 43 474024 Nov 2010 [Refereed]

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 differ...