Periodic chain of disks in a magnetic field: bulk states and edge states

Self Cite

Aharonov-Bohm ring (AB ring) is an element frequently used in nanosystems. The paper deals with wave dynamics on quantum graph consisting of AB ring coupled to a segment. It is assumed that the lengths of the edges vary in time. Variable replacement is made to come to the problem for stationary geometric graph. The obtained equation is solved using the expansion with respect to a complete system of eigenfunctions of the unperturbed self-adjoint operator for the stationary graph. The coefficients of the expansion are found as solutions of a system of differential equations numerically. The influence of the magnetic field is studied. The comparison with the case of stable geometric graph is made.

“…We put expansion (10) into equations (9) and multiply the both sides by φ differential equations for coefficients C n (t) of expansion (10):…”

Section: Quantum Graph In Magnetic Fieldmentioning

confidence: 99%

“…The Aharonov-Bohm effect was observed and studied in many physical situations (see, e.g., [2][3][4][5][6]. For such structures, quantum graph model is actively used [7][8][9][10]. It is a rather effective mathematical model allowing one to describe the spectral and transport properties of many complex physical systems [11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Wave dynamics on timedepending graph with Aharonov–Bohm ring

Eremin¹,

Grishanov²,

Nikiforov³

et al. 2018

Self Cite

“…In [17], the authors used metric graph approximation for investigation of strong variation of the viscosity and density in cylindrical domains of the small radius. And paper [18] devoted to an explicitly solvable model for periodic chain of coupled disks. In [19][20][21] IBVP for time-fractional Airy equation on the star graphs are considered via method of potentials.…”

Section: Introductionmentioning

confidence: 99%

Green's function method for time-fractional diffusion equation on the star graph with equal bonds

Sobirov¹,

Rakhimov²,

Ergashov³

2021

This work devoted to construction of the matrix-Green's functions of initial-boundary value problems for the time-fractional diffusion equation on the metric star graph with equal bonds. In the case of Dirichlet and mixed boundary conditions we constructed Green's functions explicitly. The uniqueness of the solutions of the considered problems were proved by the method of energy integrals. Some possible applications in branched nanostructures were discussed.

“…In [3] authors deal with a similar chain made up of hexagons. Two-dimensional resonators are in the focus of the papers [4][5][6], which studies spectral problems of direct and zigzag-like chains of disks, and bent chain of nanospheres respectively. Finally, chains consisting of three-dimensional resonators are presented in [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Zigzag chain model and its spectrum

Melikhova¹

2017

This work describes the development of a model using a zigzag chain of weakly-coupled ball resonators with Neumann boundary conditions. The chain is assumed to be constructed of identical resonators connected through point-like apertures. The connecting points are described by their delta-coupling with a constant intensity. The model is based on the theory of self-adjoint extensions of symmetrical operators. Due to effectively one-dimensional joints, the 3D problem can be solved with assistance from the transfer matrix approach. This allows us to study the spectrum of the physical system. In particular, it is proven that the discrete spectrum of direct zigzag chain is empty while bending deformation leads to the appearance of non-empty discrete spectrum. In addition, the continuous spectrum has band structure. With the help of asymptotic study, we obtain the dependence of the spectrum structure on the geometrical and physical parameters of the system: zigzag angle, bend angle and coupling intensity.