arXiv subjects
Publications and source records attributed to M. Harmer.
We consider a two particle system on a star graph with delta-function interaction. A complete description of the eigensolutions with real momenta is given; specifically it is shown that all eigensolutions can be written as integrals in the momentum plane of sums of products of appropriate one particle solutions.
We consider a two particle system on a star graph with $δ$-function interaction. A class of eigensolutions is described which are constructed from appropriate one particle solutions, and hence are parametrised by two momenta. These solutions include a family of solutions with discontinuous derivative on the diagonal.
The quantum graph plays the role of a solvable model for a two-dimensional network. Here fitting parameters of the quantum graph for modelling the junction is discussed, using previous results of the second author.
Hermitian symplectic spaces provide a natural framework for the extension theory of symmetric operators. Here we show that hermitian symplectic spaces may also be used to describe the solution to the factorisation problem for the scattering matrix on a graph, ie. we derive a formula for the scattering matrix of a graph in terms of the scattering matrices of its subgraphs. The solution of this problem is shown to be given by the intersection of a Lagrange plane and a coisotropic subspace which, in an appropriate hermitian symplectic space, forms a new Lagrange plane. The scattering matrix is given by a distinguished basis to the Lagrange plane. Using our construction we are also able to give a simple proof of the unitarity of the scattering matrix as well as provide a characterisation of the discrete eigenvalues embedded in the continuous spectrum.
Here we give brief account of hermitian symplectic spaces, showing that they are intimately connected to symmetric as well as self-adjoint extensions of a symmetric operator. Furthermore we find an explicit parameterisation of the Lagrange Grassmannian in terms of the unitary matrices $\U (n)$. This allows us to explicitly describe all self-adjoint boundary conditions for the Schroedinger operator on the graph in terms of a unitary matrix. We show that the asymptotics of the scattering matrix can be simply expressed in terms of this unitary matrix.
Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schroedinger operator and hence construct a Darboux transformation an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discusss the specialisation to the case of graph with trivial compact part, ie. diagonal matrix potential.
The description of electron current through a splitting is a mathematical problem of electron transport in quantum networks. For quantum networks constructed on the interface of narrow-gap semiconductors the relevant scattering problem for the multi-dimensional Schoedinger equation may be substituted by the corresponding problem on a one-dimensional linear graph with proper selfadjoint boundary conditions at the nodes. However, realistic boundary conditions for splittings have not yet been derived. Here we consider some compact domain attached to a few semi-infinite lines as a model for a quantum network. An asymptotic formula for the scattering matrix for this object is derived in terms of the properties of the compact domain. This allows us to propose designs for devices for manipulating quantum current through a splitting.
We describe inverse scattering for the matrix Schroedinger operator with general selfadjoint boundary conditions at the origin using the Marchenko equation. Our approach allows the recovery of the potential as well as the boundary conditions. It is easily specialised to inverse scattering on star-shaped graphs with boundary conditions at the node.
We derive the spectrum of the Laplace-Beltrami operator on the quotient orbifold of the non hyperbolic triangle groups.
We shown the rationality of the Taylor coefficients of the inverse of the Schwarz triangle functions for a triangle group about any vertex of the fundamental domain.
We consider a quantum graph consisting of a ring with Rashba hamiltonian and an arbitrary number of semi-infinite wires attached. We describe the scattering matrix for this system and investigate spin filtering for a three terminal device.
We propose simple conditions equivalent to the discreteness of the spectrum of the Laplace-Beltrami operator on a class of Riemannian manifolds close to warped products. For this class of manifolds we establish a relationship between discreteness of the spectrum and stochastic incompleteness.