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 $\delta$ potential and a vector potential coupled to the "loose" edges by a $\delta$ coupling. It is shown that if the lengths of the connecting edges shrink to zero and the potentials are properly scaled, the limit can yield any prescribed singular vertex coupling, and moreover, that such an approximation converges in the norm-resolvent sense.Comment: LaTeX Elsevier format, 36 pages, 1 PDF figur

We examine scale invariant Fulop-Tsutsui couplings in a quantum vertex of a general degree n. We demonstrate that essentially same scattering amplitudes as for the free coupling can be achieved for two (n − 1)-parameter Fulop-Tsutsui subfamilies if n is odd, and for three (n − 1)-parameter Fulop-Tsutsui subfamilies if n is even. We also work up an approximation scheme for a general Fulop-Tsutsui vertex, using only n δ function potentials.

Abstract. 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 finite. To the date its validity is established for numerous systems, however, it is known that quantum graphs do not comply with this law as their spectra have typically infinitely many gaps, or no gaps at all. These facts gave rise to the question about the existence of quantum graphs with the 'Bethe-Sommerfeld property', that is, featuring a nonzero finite number of gaps in the spectrum. In this paper we prove that the said property is impossible for graphs with the vertex couplings which are either scale-invariant or associated to scale-invariant ones in a particular way. On the other hand, we demonstrate that quantum graphs with a finite number of open gaps do indeed exist. We illustrate this phenomenon on an example of a rectangular lattice with a δ coupling at the vertices and a suitable irrational ratio of the edges. Our result allows to find explicitly a quantum graph with any prescribed exact number of gaps, which is the first such example to the date.

We study the scattering in a quantum star graph with a F\"ul\"op--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 the graph is strongly influenced by the potential applied on another line. On the basis of this phenomenon we design a tunable quantum band-pass spectral filter. The transmission from the input to the output line is governed by a potential added on the controlling line. The strength of the potential directly determines the passband position, which allows to control the filter in a macroscopic manner. Generalization of this concept to quantum devices with multiple controlling lines proves possible. It enables the construction of spectral filters with more controllable parameters or with more operation modes. In particular, we design a band-pass filter with independently adjustable multiple passbands. We also address the problem of the physical realization of F\"ul\"op--Tsutsui couplings and demonstrate that the couplings needed for the construction of the proposed quantum devices can be approximated by simple graphs carrying only $\delta$ potentials.Comment: 41 pages, 17 figure

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 of the lines. Adjustment of the potential strength directly controls the quantum flow from the input to the output line. This is the first example to date in which the quantum flow control is achieved by addition of an external field not on the channel itself, but on other lines connected to the channel at a vertex.PACS numbers: 03.65.Nk, 73.63.Nm The use of quantum graphs as models of quantum devices is now being widely discussed [1,2]. Quantum star graphs with n ≥ 2 lines, which are also "elementary building blocks" of any quantum graph, seem to serve particularly well for the purpose. They allow to design devices, that, although being technically simple, can have a wide scale of physical properties thanks to their large parameter spaces. One of the first applications of quantum star graphs emerged in the spectral filtering. An n = 2 star graph with the δ-interaction in its center is already usable as a high-pass filter, and similarly, a graph with the δ -interaction works as a low-pass filter. Besides of these two simple designs, the existence of an n = 3 branching filter, functionning as a high-pass/lowpass junction, has been proved [3].In principle, such a system can be controlled by a variation of the vertex parameters. But this is hard to realize in practice since it requires real-time adjustments of a nanoscale object. It would be highly desirable if the control is achieved through an external field applied onto one of the lines, preferably on lines other than those along which we want the quantum particles to propagate.In this paper we show, for the first time, that a quantum filter controllable by an external potential can be indeed designed. Besides the filter, we construct one more similar device, namely a quantum "sluice-gate" which allows to increase and decrease the quantum flux from one line to another by adjusting the external potential applied to another line. Our constructions are based on very simple star graphs with n = 3 and n = 4, respectively. The presented result may serve also as a starting point in a search for other controllable quantum device models based on quantum graphs.When a quantum particle with mechanical energy E living on a star graph comes in the vertex from the j-th line, it is scattered at the vertex into all the lines. The i-th component of the final-state wave function equals ψ ij (x) =      1

We study Schrödinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by δ-couplings with a parameter α ∈ R. 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 α = 0. We consider a "bending" deformation of the chain consisting of changing one position at a single ring and show that it gives rise to eigenvalues in the open spectral gaps. We analyze dependence of these eigenvalues on the coupling α and the "bending angle" as well as resonances of the system coming from the bending. We also discuss the behaviour of the eigenvalues and resonances at the edges of the spectral bands.

We discuss approximations of the vertex coupling on a star-shaped quantum graph of n 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 δ interactions with nonlinearly scaled couplings yields a 2n-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges one can approximate the`n +1 2´-parameter family of all time-reversal invariant couplings.

We examine scattering properties of singular vertex of degree n = 2 and n = 3, 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 δ and δ components in the connection condition between outgoing lines enables the designing of quantum spectral branch-filters.