Commutation Method Research Articles

Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies

Combining algebro-geometric methods and factorization techniques for finite difference expressions we provide a complete and self-contained treatment of all real-valued quasi-periodic finite-gap solutions of both the Toda and Kac-van Moerbeke hierarchies. In order to obtain our principal new result, the algebro-geometric finite-gap solutions of the Kac-van Moerbeke hierarchy, we employ particular commutation methods in connection with Miura-type transformations which enable us to transfer whole classes of solutions (such as finite-gap solutions) from the Toda hierarchy to its modified counterpart, the Kac-van Moerbeke hierarchy, and vice versa.

A commutator proof of the propagation of polyhomogeneity for semi-linear equations

In this paper, we show how the radial operators introduced in can be used to give a proof using commutation methods of the propagation of polyhomogeneity for conormal solutions to semi-linear equations. That conormal solutions remain conormal was first proven by Bony in, and Melrose and Ritter later showed in that the result could be proven by using testing and commutation methods. Ranch and Reed in later showed that for a suitable notion of polyhomogeneity that polyhomogeneity was also preserved for first order systems. This was later reprover and generalized by Piriou in using symbolic methods. 14 refs.

Acoustic noise cancellation techniques for switched reluctance drives

The levels of acoustic noise emitted by switched reluctance motor (SRM) drives are noted as being above those of competing variable speed drives. In other respects, the SRM drive offers distinct advantages. In a previous paper, the authors described a new technique for the cancellation of radial vibrations of the stator. This new technique was limited to power converters offering a zero voltage freewheel path. This paper describes further development of the technique which allows noise cancellation to be achieved with all types of power converter. The techniques disclosed in this paper include a smoothing method, a three-stage commutation method for converters with one switch per phase and a noise cancellation technique suitable for SRM drives incorporating an extended zero-voltage freewheel period for improved efficiency. The techniques are all compared, with experimental and simulated results.

One-dimensional scattering theory for quantum systems with nontrivial spatial asymptotics

We provide a general framework of stationary scattering theory for one-dimensional quantum systems with nontrivial spatial asymptotics. As a byproduct we characterize reflectionless potentials in terms of spectral multiplicities and properties of the diagonal Green's function of the underlying Shrödinger operator. Moreover, we prove that single (Crum-Darboux) and double commutation methods to insert eigenvalues into spectral gaps of a given background Shrödinger operator produce reflectionless potentials (i.e., solitons) if and only if the background potential is reflectionless. Possible applications of our formalism include impurity (defect) scattering in (half-) crystals and charge transport in mesoscopic quantum-interference devices.

Commutation Methods for Jacobi Operators

We offer two methods of inserting eigenvalues into spectral gaps of a given background Jacobi operator: The single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi-bounded background Jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps. Moreover, we prove unitary equivalence of the commuted operators, restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues, with the original background operator. In addition we compute the (matrix-valued) Weylm-functions of the commuted operator in terms of the background Weylm-functions. Finally we show how to iterate the above methods and give explicit formulas for various quantities (such as eigenfunctions and spectra) of the iterated operators in terms of the corresponding background quantities and scattering matrix. Concrete applications include an explicit realization of the isospectral torus for algebro-geometric finite-gap Jacobi operators and theN-soliton solutions of the Toda and Kac–van Moerbeke lattice equations with respect to arbitrary background solutions.

On the double commutation method

We provide a complete spectral characterization of the double commutation method for general Sturm-Liouville operators which inserts any finite number of prescribed eigenvalues into spectral gaps of a given background operator. Moreover, we explicitly determine the transformation operator which links the background operator to its doubly commuted version (resulting in extensions and considerably simplified proofs of spectral results even for the special case of Schrodinger-type operators).

Completely transparent potentials for the Schrödinger equation

Constructions of local potentials with positive eigenvalues for the one-particle Schr\odinger equation are presented. The complete transparency of these potentials, manifesting itself in vanishing scattering phases for potentials over the semiaxis and vanishing reflection coefficients with transmission coefficients equal to one for the full axis, is shown. The results derived here via Darboux transformations are compared with those obtained in similar approaches such as the double commutation method, the Gel'fand-Levitan formalism, or supersymmetric quantum mechanics. An application within the theory of nonlinear waves is indicated.

A study on dynamic performance of closed-loop commuted stepping motor to improve precision motion control

To achieve precision movements, a motor with very low speed ripple and torque ripple is required. A motor with more poles will normally give higher ripple frequency and hence lower ripple magnitude. The hybrid stepping motor when commuted in closed-loop mode will behave in the form of a multi-pole brushless DC servo motor with each step similar to one pole. To reduce torque ripple further, the commutation of the coils requires careful adjustment of the lead angle in order to compensate for the time constant of the coils at different speeds. This article establishes an accurate dynamic model for the closed-loop commuted hybrid-type stepping motor, which takes into account the drive circuit and the effect of magnetic saturation. Simulations have been performed on the model using the stiff system numerical method for higher accuracy. The results show a good match between the numerical solution and the actual measurement. The dynamic model developed can be used to predict speed ripple of the motor system under various commutation methods. This provides a means of minimizing speed ripple by using software to improve the accuracy of motion control systems on high-precision machines.

Absolute continuity of the integrated density of states of the quantum Lorentz gas for a class of repulsive potentials

For a class of repulsive potentials, for instance of the type phi (x)= mod x mod - kappa (1+x2)- lambda ( kappa , lambda positive, in a certain range) in the random Schrodinger operators H( omega )=p2+V (x, omega )=P2+ Sigma j phi (x-xj), acting in L2 (Rd), with Poisson distributed xjs (the quantum Lorentz gas), we show that the integrated density of states N(E) is absolutely continuous for E> zeta rho ( phi ). Here is ( phi ) the integral of phi over Rd, rho the averaged density of points xj and zeta >0 depends on phi and d. In the above example, zeta =(d/ kappa )2. Our method makes use of a Fock space representation for the Poisson random system, recently developed by Maassen and the author (1993). Within this Fock space formalism the Mourre commutator method is then employed to obtain the announced result.

A Complete Spectral Characterization of the Double Commutation Method

We provide a complete spectral characterization of the double commutation method which introduces eigenvalues into arbitrary spectral gaps of a given background Schrödinger operator in L 2( R ) respectively in L 2((0, ∞)). Our main result proves unitary equivalence of the doubly commuted operator, restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues, with the original background operator. Moreover, we explicitly compute the (matrix) spectral function of the doubly commuted operator in terms of the background spectral function.

Scattering Theory for $N$-Particle Systems with Stark Effect: Asymptotic Completeness

The present paper is a continuation to the work [15] where we have proved the non-existence of bound states and the principle of limiting absorption for /V-particle Stark Hamiltonians. We here study the problem of asymptotic completeness of wave operators. The asymptotic completeness for Nparticle quantum systems without uniform electric fields was first proved by Sigal-Soffer [12] and after that remarkable work, alternative proofs have also been given by several authors [2, 7, 13, 17]. We use the local commutator method initiated by Mourre [9] to prove the asymptotic completeness for Nparticle systems with uniform electric fields. The proof is, in principle, based on the same idea as developed in the above works [2, 7, 12, 13, 17] for the case without electric fields. We analyse the propagation properties in the configuration space for solutions to the Schrodinger equation. But the phase space analysis is not required, because charged particles are scattered along only one direction (direction of a given uniform electric field). We shall formulate the problem precisely, fixing several basic notations employed in many-particle scattering theory. We consider a system of /Yparticles moving in a constant electric field £e/2, <?^0. We denote by in.,, e-, and r^e/2, 1^/^A/, the mass, charge and position vector of the /-th particle, respectively. Then, for such a system, the total energy Hamiltonian takes the following form:

Principle and Theoretical Performance of the Saturable Transformer Fluxpump with a Resistive Commutation Method.

A rectifier fluxpump (superconducting rectifier) is one of power source units for a superconducting magnet. This apparatus converts an AC current having a small amplitude into a large DC current suitable for the superconducting magnet using a superconducting transformer and two superconducting switches. A typical conventional rectifier fluxpump with the superconducting transformer by using an air-core has such defects that the pumping flux is small and decreases as charging current for the superconducting magnet increases. We noted the fact that a magnetization curve of magnetic core of the superconducting transformer is related with pumping flux of the rectifier fluxpump and studied a new-type rectifier fluxpump with the superconducting transformer by using a saturable iron-core, namely, saturable transformer fluxpump.In this paper, we consider principle and theoretical performance of the saturable transformer fluxpump with a resistive commutation method. The analysis shows the high pumping flux, constant rates for increasing the charging current and high efficiency.

Use of inter-electrode commutation for elimination of non-Hall voltages in inhomogeneous Hall generators

The non-Hall voltages in Hall generators made of electrically inhomogeneous cross-shaped slabs are considered. It is shown that these voltages can be eliminated with an inter-electrode commutation (IEC) method. In the result the output voltage of a Hall generator becomes cleared of the perturbing voltages. IEC can be particularly useful in application to thin-film Hall generators, as they usually have electrical inhomogeneities. With IEC the thin-film Hall generators can be used for precise measurements or stabilization of magnetic fields.

The state of the art of the BKL commutator method

Boson mappings have been extensively used to link the interacting boson mapping (IBM) to the shell model and hence, give an underlying justification to the IBM. Here we test one of this mappings called BKL (Bonatsos-Klein-Li) in a single j-shell, in a single j-shell with protons and neutrons and in two non-degenerate j-shells.

Spectral analysis for optical fibres and stratifled fluids II: Absence of Eigenvalues

The operator Ĥ appears in the study of the propagation of acoustical waves in stratified fluids , as well as of electromagnetic waves in layered dielectric media , and in optical fibres . The density and signal speed are (possibly long range) perturbations of a density and speed , . Positive commutator methods along with Virial techniques are used to show that in many cases of physical interest the point spectrum of Ĥ is empty. More generally, eigenfunctions of Ĥ if they exist, must decay faster than .

Limiting amplitude principle for acoustic propagators in perturbed stratified fluids

By use of the commutator method, we study the resolvent estimate at low frequencies of acoustic operators − c( x) 2 Δ in perturbed stratified fluids and prove the principle of limiting amplitude for such operators.

The commutator method for form-relatively compact perturbations

We present a variant of the conjugate operator method which can be used when the group generated by the conjugate operator leaves invariant only the form domain of the Hamiltonian. As an example, we get detailed spectral properties and a large class of locally smooth operators for two-body Schrodinger Hamiltonians with form-relatively compact potentials.

Spectral analysis for optical fibres and stratified fluids I: The limiting absorption principle

The wave equation (∂ t 2+ H ̂ )Ψ=0 with H ̂ =−c 2(x,y)ρ(x,y)▽ z·ρ(x,y) −1▽ z, z=(x,y),xϵR k,yϵR m , describes the propagation of acoustical waves in stratified fluids ( m = 1, k = 2), as well as electromagnetic waves in layered dielectric media ( m = 1, k = 2), and in optical fibres ( m = 2, k = 1,and p( x, y) = 1). The density p( x, y) and signal speed c( x, y) are (possibly long range) perturbations of a density p 0( y) and speed c 0( y), i.e., ( p( x, y) − p 0( y)), ( c( x, y) − c 0( y)) → 0 as ¦z¦→ ∞. Positive commutator methods are used to show that the spectrum of H ̂ is absolutely continuous, except possibly for a sequence of isolated eigenvalues of finite multiplicity that can accumulate only at zero and ∞. Away from those eigenvalues a limiting absorption principle for H ̂ is established: it is proven that the resolvent ( H ̂ − k) −1 of H ̂ has a norm limit as an operator between L 2(R k+m,(1+z 2) α 2 (c 2ρ) −1 d nz) and L 2(R k+m, (1+z 2) −α 2 (c 2ρ) −1 d nz) for α > 1 2 as kϵC\\ R + approaches the real axis either from above or from below.

Commutation Methods Applied to the mKdV-Equation

An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., $N = 1$ supersymmetry) underlying Miura’s transformation that links solutions of the two evolution equations.

Commutation methods applied to the mKdV-equation

An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., N = 1 supersymmetry) underlying Miura's transformation that links solutions of the two evolution equations.