Vadim Kostrykin

Kirchhoff's rule for quantum wires

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 Kirchhoffs 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.

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 ``Kirchhoffs 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

Quantum Wires with Magnetic Fluxes

ntimes n

The generalized star product and the factorization of scattering matrices on graphs

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.

CONCAVITY PROPERTIES OF KREIN’S SPECTRAL SHIFT FUNCTION

Abstract:u2002In 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.

Cluster Properties of One Particle Schrödinger Operators. II

In this article we continue our analysis of Schrodinger operators on arbitrary graphs given as certain Laplace operators. In the present article we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs. We perform a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discussed.

Microholes in zirconia-coated Ni-superalloys for transpiration cooling of turbine blades

We prove that the integrated Krein’s spectral shift function for one particle Schrodinger operators in R3 is concave with respect to the perturbation potential. The proof is given by showing that the spectral shift function is the limit in the distributional sense of the difference of the counting functions for the given Hamiltonian and the free Hamiltonian in a finite domain Λ with Dirichlet boundary conditions when Λ→∞.

Picosecond-laser-pulse-induced heat and mass transfer

We continue the study of cluster properties of spectral and scattering characteristics of Schrodinger operators with potentials given as a sum of two wells, begun in our preceding article [Rev. Math. Phys. 6 (1994) 833–853] and where we determined the leading behaviour of the spectral shift function and the scattering amplitude as the separation of the wells tends to infinity. In this article we determine the explicit form of the subleading contributions, which in particular show strong oscillatory behaviour. Also we apply our methods to the critical and subcritical double well problems.

Determination of the Scattering Amplitudes of Schrödinger Operators from the Cross-Sections, a New Approach

Drillings in zirconia coated Ni-superalloys is done by melt extraction with pulsed laser radiation provided by a Nd:YAG slab laser with microsecond pulse duration. This laser system distinguishes itself by a high beam quality and offers the possibility to investigate drilling of holes with a diameter of 200 micrometer by percussion drilling and trepanning. The quality of drilled holes, e.g. the heat affected zone (HAZ), the recast layer and the conicality, are presented. During drilling different process gases are used. The results in drilling velocities, melt thickness and chemical composition of the melting zone are shown for oxygen, argon and nitrogen by SEM and EDX. A numerical simulation of the trepanning process will be presented. The different time scales of the contributing physical processes related, for example, to the small melt film layer during trepanning are described. A coating is distributed on the multilayer system to protect the blade from recast. Aim of the investigation is the production of holes in a multilayer system, consisting of CMSX-4, VPS-MCrAlY and EB-PVD-zirconia. With this used laser system inclined holes up to 60 degrees in this layer system can be drilled. No recast layer and no spalling of the zirconia-layer are observed.

Ionization of atoms and molecules by short, strong laser pulses

The interaction of picosecond and sub-picosecond laser pulses with metals is investigated both, theoretically and experimentally. Analyzing the Boltzmann equation for electrons and phonons the hyperbolic two-temperature model of heat conduction in metals is obtained. In particular the parameter range for which the hyperbolic effects are significant is analyzed. For calculations a numerical algorithm based on the method of lines is developed. Experimentally laser pulses with a duration of 40 ps are used to remove thin metal films. The removal process is analyzed by pump and probe measurements with the time resolution of 40 ps. The single-shot removal threshold and the removal rate per pulse are determined for copper. By this technique the existence and the propagation of shock waves in the ambient atmosphere induced by the removal process are detected. Theoretical calculations are compared with experiments and the results from the literature.