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MATHEMATICAL MODEL OF RESONATOR CHAINS IN EXTERNAL MAGNETIC FIELD

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 Subject of Research.The paper considers a spectral problem for chains of weakly coupled ball-shaped resonators under the action of an external magnetic field. Two basic geometries are studied composed of identical resonators: the chain with a single kink and the one with symmetric branching. The vector of magnetic field induction is assumed to be perpendicular to the plane including the centers of resonators that are forming the chain. The delta-like potentials with the same intensity are applied to all contact points. Method. The mathematical model of the chain consisting of identical resonators interacting with each other via a single point is constructed in the framework of the theory of self-adjoint extensions of symmetric operators. The modification of Neumann formulas was used for adjoint operator description. Both systems have a periodicity in parts and the model itself is constructed in such way that it becomes possible to use the transfer matrix approach to obtain the spectrum of the model Hamiltonian. Main Results. The structure of the spectrum for both chains is analytically described; namely, all equations and inequalities are obtained, which give the possibility to find the bands of the continuous spectrum, as well as the energy values ​​related to the discrete spectrum. Numerical simulation clearly demonstrates the structure of the continuous spectrum of the systems depending on the model parameters. Practical Relevance. Since the model has different variable parameters, it can be used to construct real systems that have certain spectral properties. The geometries considered in this paper are the basic elements for the construction of more complex systems.

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