Kirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers

In this article we continue our investigations of one particle quantum scattering theory for Schroedinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In ``Kirchhoff's rule for quantum wires'' [J. Phys. A: Math. Gen. 32, 595 - 630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary $n\times n$ matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian.Comment: 16 page

Quantum wires with magnetic fluxes

In the present article magnetic Laplacians on a graph are analyzed. We provide a complete description of the set of all operators which can be obtained from a given self-adjoint Laplacian by perturbing it by magnetic fields. In particular, it is shown that generically this set is isomorphic to a torus. We also describe the conditions under which the operator is unambiguously (up to unitary equivalence) defined by prescribing the magnetic fluxes through all loops of the graph

Statistical ensembles and density of states

We propose a definition of microcanonical and canonical statistical ensembles based on the concept of density of states. This definition applies both to the classical and the quantum case. For the microcanonical case this allows for a definition of a temperature and its fluctuation, which might be useful in the theory of mesoscopic systems. In the quantum case the concept of density of states applies to one-particle Schroedinger operators, in particular to operators with a periodic potential or to random Anderson type models. In the case of periodic potentials we show that for the resulting $n$-particle system the density of states is $[(n-1)/2]$ times differentiable, such that like for classical microcanonical ensembles a (positive) temperature may be defined whenever $n\geq 5$. We expect that a similar result should also hold for Anderson type models. We also provide the first terms in asymptotic expansions of thermodynamic quantities at large energies for the microcanonical ensemble and at large temperatures for the canonical ensemble. A comparison shows that then both formulations asymptotically give the same results.Comment: MSC-class: 47A40, 82B10 (Primary) 82B05, 82D20 (Secondary

Current Densities in Density Functional Theory

It is well known that any given density rho(x)can be realized by a determinantal wave function for N particles. The question addressed here is whether any given density rho(x) and current density j(x) can be simultaneously realized by a (finite kinetic energy) determinantal wave function. In case the velocity field v(x) =j(x)/rho(x) is curl free, we provide a solution for all N, and we provide an explicit upper bound for the energy. If the velocity field is not curl free, there is a finite energy solution for all N\geq 4, but we do not provide an explicit energy bound in this case. For N=2 we provide an example of a non curl free velocity field for which there is a solution, and an example for which there is no solution. The case $N=3 with a non curl free velocity field is left open.Comment: 21 pages, latex, reference adde

Energy Transfer in Scattering by Rotating Potentials

Quantum mechanical scattering theory is studied for time-dependent Schroedinger operators, in particular for particles in a rotating potential. Under various assumptions about the decay rate at infinity we show uniform boundedness in time for the kinetic energy of scattering states, existence and completeness of wave operators, and existence of a conserved quantity under scattering. In a simple model we determine the energy transfered to a particle by a collision with a rotating blade.Comment: To appear in the Proceedings of the Workshop on Spectral and Inverse Spectral Problems for Schroedinger Operators, Goa, India, December 14 - 20, 200