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Nanosystems Physics Chemistry Mathematics

A model of an electron in a quantum graph interacting with a two-level system

Anton A. Boitsev, I. Yu. Popov

Published Apr 27, 2019• 1 citation
Publisher

Abstract

A model of an electron in a quantum graph interacting with a two-level system is considered.The operator describing the model has the form of sum of tensor products.Self-adjoint extensions and a scattering matrix are written in terms of a boundary triplet, corresponding to the considered symmetric operator.Diagrams of reflection are calculated and numerical results are discussed .

DOI
10.17586/2220-8054-2019-10-2-131-140

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A model of an electron in a quantum graph interacting with a two-level system is considered.The operator describing the model has the form of sum of tensor products.Self-adjoint extensions and a scattering matrix are written in terms of a boundary triplet, corresponding to the considered symmetric operator.Diagrams of reflection are calculated and numerical results are discussed .
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Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
It is shown that the overlap integral between two ground-state wave functions for the Hamiltonian of the interacting electron system with different local potentials can be written generally in terms of the S-matrix describing potential scattering. Here, the local potential is considered to be a general inseparable one and the Green functions and self-energies are treated as operaors with off-diagonal elements. This orthogonality theorem expressed in a general form is applied to some special cases.
We formulate and discuss one-particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S -matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E>0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low-energy behaviour of one theory gives the high-energy behaviour of the transformed theory. Finally, we provide a composition rule by which the on-shell S -matrix of a graph is factorizable in terms of the S -matrices of its subgraphs. All proofs use only known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitian symplectic forms.