Parameter regime of a resonance quantum switch

Boundary condition at the junction

“…This is a perturbation on the continuous spectrum, so the convergence of the corresponding series can't be estimated in spectral terms of self-adjoint operators. In [13] we suggested a modified analytic perturbation procedure based on introduction of an Intermediate Hamiltonian obtained via appropriate splitting, see [15] of L.…”

“…We, from [13] boundary condition similar to (2) for any junction from [13] and interpret the corresponding free parameter P 0 in terms of standing waves on the corresponding vertex domain.…”

Resonance scattering across the superlattice barrier and the dimensional quantization

“…2 Notice, that I. Prigogine in 1973 formulated the hypothesis on the validity of the Poincare two-step algorithm of analytic perturbation procedure for quantum problems, see [27], but could not prove it, because selected an incorrect anzsatz was selected for the corresponding Intermediate operator. The hypothesis was later proved for Quantum Networks based on the correct ansatz, [22], for the intermediate operator, presented in the form of zero-range model with an inner structure, constructed with use of Lagrangian technique of operator extension procedure [23] with differentiation in an outgoing direction at the node. The boundary form of the inner Hamiltonian A is calculated in a special representation of the boundary form J int (u, v), constructed based on an orthogonal basis {e l } ⊂ N i :…”

“…Later probably used another approach to the problem of extending of a symmetric operator to the corresponding self-adjoint, which yields a convenient formula for the scattered waves, see [19]. Though the conventional proof of self-adjointness of the Laplacian under the above boundary conditions was proposed 25 years later, see [20], the approach to operator extension based on the boundary form proved to be extremely efficient, see for instance [12,13,21] 1 In 1970's, it was modified by introducing the inner structure into zero-range potential, see [12,13], that allowed consideration of resonance interaction, which allows admission of fitting based on asymptotics Dirichlet-to-Neumann map of the corresponding unperturbed problem, see [22]. This approach allows one to develop an analytic perturbation technique for embedded eigenvalues, based on two step analytic perturbation procedure -a quantum Jump-Start analog of the corresponding classical techniques developed by Poincare [23] and, in particular, to propose a convenient solvable model for Quantum Networks, supplied with inner structure on the nodes, see [24] 2 .…”

Spin filtering on a ring with the Rashba Hamiltonian

“…We also note that projection type boundary conditions, also referred to in the degree two case as Fülöp-Tsutsui point interactions, are important in the study of 'quantum chaotic' behavior in quantum graphs [12] due to their scale invariance. Given a two or three dimensional quantum network, the problem of deriving appropriate boundary conditions at the vertices of an approximating one dimensional quantum graph is an active area of research (see [2,7,11]) as is the 'inverse problem' of constructing sequences of graphs with regular potentials so that in the limit we observe a chosen boundary condition from the whole U (n) parameter space [6,3].…”