Two interacting particles in deformed nanolayer: discrete spectrum and particle storage

Abstract

The problem of particle storage in a nanolayered structure is considered. Local perturbations of the nanolayers can lead to the appearance of eigenvalues of the corresponding one-particle Hamiltonian. To study the particle storage it is necessary to deal with a multi-particle problem. The Hartree method and the finite element method are used. The discrete spectrum of the Hamiltonian of two interacting particles is considered. Two different types of the perturbation are treated: deformation of the layer boundary and a small window in a wall between two layers. The relation between the system parameters (interaction intensity–waveguide deformation) ensuring the existence of a non-empty discrete spectrum is studied. Comparison of particle storage efficiencies in these two cases is made.

On the existence of point spectrum for branching strips quantum graph

“…Note that the system (12) coincides with (14) for the matrix P + and the system (13)-with system (14) for the matrix P − (previously, it was proved that the eigenvalues of matrices P and P ± are equal). So, we found four independent solutions of the original system (11) and because of the system (11) has exactly four solutions, other solutions do not exist. Thus, the spectral analysis of the matrix P can be replaced by spectral analysis of the matrices P ± , q.e.d.…”

“…[6][7][8][10][11][12] To describe physical properties of nanosystems one should not ignore the information about the bound states. Quantum graphs with a hexagonal lattice are a special case of quantum graphs, but they allow us to investigate many nanostructures.…”

Asymptotic and numerical analysis of slowly varying two-dimensional quantum waveguides

“…Most of the literature on quantum waveguides is focussed on the spectral and scattering properties of the quantum particles subjected to interaction conditions [7,[18][19][20][21][22][23][24][25][26][27][28]. Moreover, the influence of the geometry of the confinement region Π on the spectral properties of the waveguide has also been studied with emphasis on the curvature-induced bound states [25,27,[29][30][31], although other aspects have been considered [27,[32][33][34].…”

Window-coupled nanolayers: window shape influence on one-particle and two-particle eigenstates

“…We make numerical calculations, with the application of Hartree-Fock estimation method (for the Hartree approximation accuracy, see [28]). As for previous studies of the multi-particle problems in deformed waveguides, see, e.g., [23][24][25][26][27]. We are concerned with the following two questions: how the energies of bound states and number/position of nodal domains are dependent on shape of a window and what is the relation between one and two-particle cases.…”