Comment on "Optimized matrix inversion technique for the T-matrix method"

The matrix inversion technique for the T-matrix method recently suggested by Petrov et al. [Opt. Lett.32, 1168 (2007)] is not new. Their way of implementing the technique, in which Strassen's matrix multiplication algorithm is not adopted, does not save computation time when the LU factorization technique of matrix computation theory is used.     

Optimized matrix inversion technique for the T-matrix method

We suggest a new approach to calculate the inverse matrix in scattering calculations using the T-matrix method. Instead of inversion of the full matrix, we suggest the inversion of two matrices, each of which contains half the number of rows. This approach allows significant time savings and a noticeable increase of the precision of scattering calculations due to fewer arithmetical operations. An iterative method can be applied to matrices whose dimension is also divisible by factors of 2, which can further increase the time savings and accuracy.     

The mathematical structure of the Kondo resonance

TheT-matrix of the Nagaoka-Suhl theory for the scattering of conduction electrons from a magnetic impurity is discussed in the weak coupling limit. A law of corresponding states is established where the Kondo temperature appears as a scale factor. Some characteristic features of theT-matrix are listed which are to be used in a subsequent calculation of the spin correlation function.     

T-matrix analysis of biexcitonic correlations in the nonlinear optical response of semiconductor quantum wells

A detailed numerical analysis of exciton-exciton interactions in semiconductor quantum wells is presented. The theory is based on the dynamics-controlled truncation formalism and evaluated for the case of resonant excitation of 1s-heavy-hole excitons. It is formulated in terms of standard concepts of scattering theory, such as the forward-scattering amplitude (or T-matrix). The numerical diagonalization of the exciton-exciton interaction matrix in the 1s-approximation yields the excitonic T-matrix. We discuss the role of the direct and exchange interaction in the effective two-exciton Hamiltonian, which determines the T-matrix, evaluated within the 1s-subspace, and also analyze the effects of the excitonic wave function overlap matrix. Inclusion of the latter is shown to effectively prevent the 1s-approximation from making the Hamiltonian non-hermitian, but a critical discussion shows that other artefacts may be avoided by not including the overlap matrix. We also present a detailed analysis of the correspondence between the excitonic T-matrix in the 1s-approximation and the well-known T-matrix governing two-particle interactions in two dimensional systems via short-range potentials. Received 3 August 2001 and Received in final form 26 December 2001     

The T-matrix for particle with arbitrary permittivity tensor and parallelization of the computational code

Using the T-matrix approach electromagnetic scattering by a non-axisymmetric particle with an arbitrary permittivity tensor is studied. The electromagnetic fields inside the scatterer are expressed by a system of quasi-spherical vector wave functions which are derived by the use of inverse Fourier transform. Using this expansion a solution of the light scattering problem in the framework of the null-field method is obtained. The implemented T-matrix program is parallelized using both OpenMP and MPI parallel paradigms. Numerical scattering results for anisotropic ellipsoids are presented.     

Microscopic R-matrix theory in a generator coordinate basis: (III). Multi-channel scattering

The microscopic R-matrix theory presented in two previous papers is extended to the multichannel scattering case. An antisymmetrized wave function is built in the two-centre harmonic oscillator model. This wave function is proved to be equivalent to the resonating group one even if the nuclei have non-zero spin. The method only requires the calculation of numerical values of matrix elements between Slater determinants. The microscopic R-matrix theory may be applied to study both reactions and inelastic scattering.     

Renormalized Perturbation Expansion in Kinetic Theory

Ladder summation techniques are applied to the recently developed c-number diagram expansion for kinetic equations describing the relaxation of quantum fluids. By way of the resummation, the kinetic equations are reformulated in terms of a T-matrix which describes the scattering of two particles influenced by the remaining particles of the system via the particle statistics (Bose or Fermi). At sufficiently high temperatures (Maxwell-Boltzmann statistics), the T-matrix introduced coincides with the usual T-matrix of conventional two-body scattering theory in free space. In low-temperature Fermi systems, the T-matrix differs from the reaction matrix of the Brueckner-Goldstone theory because the “healing” property of the two-body wave function does not obtain.     

A problem of stability in phase shift analysis

In this paper we present a method which avoids—in RGM calculations—ex0licit evaluations of overlap kernels in configuration space. Phase shifts are directly calculated by the use ofT-matrix formalism, without resorting to numerical evaluations of non-local Schrödinger equations. Different two body potentials can be directly compared within the very same RGM calculation. As an example, we evaluateS waveI=1,0K?N phase shifts for three different sorts of quark-quark potentials: The ones used by Bhaduri et al. [3], Bender et al. [4], and finally by Campbell and Robson [8]. These phase shifts are shown to depend sharply on the details of the adopted interquark force, therefore distinguishing among potentials which otherwise fare equally well for spectroscopic calculations. None of the above mentioned potentials suffice to explain the experimentally observedS waveK?N phase shifts.     

The Gell-Mann-Goldberger formula for long-range potential scattering

A derivation of the Gell-Mann-Goldberger (GG) formula and cut-off versions of this formula for the T-matrix involving long-range potentials is given. The derivation is based on the time-dependent and recently developed stationary formalisms for scattering via long-range potentials. A stationary S-operator expression for two-body Coulomb-like scattering is derived. Using the well-known expression for the off-energy-shell “T-matrix” for a pure Coulomb potential the energy-shell limit of this stationary expression is shown to yield the pure Coulomb scattering amplitude. A proof of the convergence of the perturbation series corresponding to the Gell-Mann-Goldberger formula for the two-body Coulomb-like T-matrix is given.     

The dependence of the photodisintegration cross section on the off-shell T-matrix

Nucleon-nucleon scattering phase shifts determine the diagonal element of the transition matrix. The off-diagonal elements are not completely arbitrary but have conditions imposed on them by the range and the tail of the potential. Electromagnetic interaction can also be used to place restrictions on the off-diagonal elements. We find that the cross section of the deuteron photodisintegration is sensitive to the off-shell transition matrix. The integrated cross section can be varied by as much as 30 % or more, and the matrix element for the El transition by a factor of 2. While the matrix element for the photodisintegration depends on the off-shell elements of the T-matrix, it cannot be used to discriminate between alternative off-shell T-matrices. We have constructed classes of different off-shell T-matrices, which produce identical photo-disintegration cross sections and other two-body scattering and bound-state properties.     

Coupled-channel R-matrix method on a Lagrange mesh

The coupled-channel R-matrix method on a Lagrange mesh is a very simple approximation of the R-matrix method with a basis. The mesh points are zeros of shifted Legendre polynomials. Bound-state energies and scattering matrices are easily calculated with small numbers of potential values at mesh points. A test with an exactly solvable two-channel potential provides an excellent accuracy over a broad energy range with only 30 mesh points. The efficiency of the method is illustrated for a single channel on α + α scattering and for two channels on the deuteron ground-state energy and on nucleon-nucleon scattering.     

A molecular T-matrix approach to calculating Low-Energy Electron Diffraction intensities for ordered molecular adsorbates

We present a novel approach to calculating Low-Energy Electron Diffraction (LEED) intensities for ordered molecular adsorbates. First, the intra-molecular multiple scattering is computed to obtain a non-diagonal molecular T-matrix. This is then used to represent the entire molecule as a single scattering object in a conventional LEED calculation, where the Layer Doubling technique is applied to assemble the different layers, including the molecular ones. A detailed comparison with conventional layer-type LEED calculations is provided to ascertain the accuracy of this scheme of calculation. Advantages of this scheme for problems involving ordered arrays of molecules adsorbed on surfaces are discussed.     

Derivation of extended boundary condition method from general definition of -matrix elements and Lippman–Schwinger equation for transition operator

Waterman's surface-integral expressions for the T-matrix elements are derived on the basis of the quantum-mechanical potential scattering approach in electromagnetic scattering problem. We use general definition of the elements of the T-matrix as the matrix elements of the dyadic transition operator and Lippman–Schwinger volume integral equation for the dyadic transition operator. The interrelations of the Q- and Waterman's T-matrix with the transition operator are shown.     

Summation of partial waves for large-angle scattering

For large-angle elastic scattering different methods of summing partial wave amplitudes are investigated for their accuracy and simplicity of computation. It is found that among the approximations considered, the method of expanding the T-matrix in terms of the weighted orthogonal polynomials proposed by Brysk is the most accurate way of calculating the scattering amplitude in the backward direction. If the two-particle interaction is assumed to be a Yukawa potential, then the lth partial sum of the T-matrix with the weighted polynomials can be expressed as the lth partial sum with the Legendre polynomials and a correction term which depends on the phase shift for the lth partial wave.     

Off-energy-shell continuation of the πN transition matrix element

General relationships of the fully off-shell T-matrix element for the central forces are derived. The problem of finding the fully off-shell T-matrix element can be reduced to finding a real two-parameter function φ(p, k). It can be shown in the case of no bound state that if φ(p, k) is separable, the whole problem becomes equivalent to the inverse scattering problem. An approximate method to include inelastic effects is proposed and the πN off-shell T-matrix element is calculated.     

Light scattering by a finite cylinder containing a spherical cavity using Sh-matrices

We use the Sh-matrix formalism that contains the shape-dependent parameters of the T-matrix to derive an analytical solution for the light scattering from a finite cylinder containing a spherical cavity. The integral expressions for the Sh-matrix elements are simpler than those of the T-matrix elements and the case of a sphere embedded in a finite cylinder these integrals can be solved analytically.     

The structure of Heesch groups and its relation to material property tensors

The magnetic crystal point groups (Heesch groups) are classified according to their structure with respect to the three inversion operations: space, time, and total inversion. Accordingly the tensors are classified by the irreducible representations of the full inversion group. The groups and tensors are considered under the action of the elements A_i of the group of automorphisms of the full inversion group. The following correspondence theorem is proved: The matrix form of the tensor representation T of the group G coincides with the matrix form of the representation A_iT of the group A_iG. The theorem gives a clear explanation of the so-called “magic numbers” and provides a suitable short cut for the calculation and tabulation of material property tensors.     

On the T-matrix formula for multichannel coulomb scattering

It is pointed out that in scattering theory for Coulomb like forces the conventional expression for the off-energy-shell T-matrix does not lead to the correct on-energy-shell T-matrix. An alternative expression giving the correct on-energy shell limit is discussed.