Determinantal Method Research Articles

On analytical continuation in the determinantal method

On analytical continuation in the determinantal method

Generalization of the Determinantal Method to Continuous Channels

The procedure by which all elements of the S matrix in multichannel scattering problems are expressed in terms of the Fredholm determinant of the Lippmann-Schwinger equation, is generalized to the case of ``continuous channels.'' This is a preliminary step toward its generalization to three (or more)-particle problems, in which the possibility of ``ionizing'' an initial bound state always introduces such a continuum. It is found that, whereas in the case of discrete channels the necessary functions can be obtained from the Fredholm determinant either by analytic continuation in the total energy, or by a ``substitution rule,'' for continuous channels only the latter procedure works, and it is not equivalent to an analytic continuation.

On the theory of photo- and electroproduction of theN

Multipole dispersion relations and the Fredholm determinantal method are applied to the calculation of the transverse part of the γN N * vertex, using relativistic kinematics. The unphysical cuts of the multipole amplitudes have been approximated by the second-order and fourth-order (box) diagrams containing nucleon and pion exchange. No cut-off is necessary. The results for the magnetic-dipole amplitude agree, within the theoretical errors, with those of the static theory, while the electric quadrupole turns out to be resonant too. The behaviour of the transition amplitudes for large momentum transfer is discussed.

Application of an Approximate Solution of Partial-Wave Dispersion Relations to Yukawa Potential Scattering

The time-reversal symmetrization of the multichannel scattering amplitude proposed by Fulton and Shaw is used to construct the amplitude for nonrelativistic single-channel Yukawa potential scattering. It provides a modified determinantal method for the solution of the $\frac{N}{D}$ equations. This amplitude is compared to the exact solution of the Schr\"odinger equation. It is found that the scattering lengths predicted by this method are qualitatively the same as those predicted by the full $\frac{N}{D}$ equations and significantly better than the results of the determinantal method. The computational simplicity of the determinantal method has been retained and combined with the accuracy of the full $\frac{N}{D}$ solution, which, with first-order Born approximation, gives quite a reliable picture of the qualitative features of the scattering.

Application of a modified determinantal method to Yukawa potential scattering

Using the modified determinantal method of Nath and Srivastava, theS, P andD wave effective-range values are calculated for the Yukawa potential. The results are then compared with the effective-range values predicted by the usual determinantal method and are found to be superior.

Dynamics of a Singlet and an Octet ofD32Meson-Baryon Composite States

The ${D}_{\frac{3}{2}}$ meson-baryon resonances are investigated on the basis of a multichannel Bethe-Salpeter equation in the unitary symmetry limit. The driving mechanism consists of a box diagram which provides strongly attractive $D$-state forces corresponding to the unitary singlet and octet states. A reduction of the Bethe-Salpeter equation to a single-dimensional relativistic integral equation which incorporates two-body unitarity is effected along the lines of a method of Blankenbecler and Sugar. The equation is solved by the determinantal method. The model accounts for a singlet and gamma octet of composite states. The paper contains some relevant results on the application of the methods of parity and helicity amplitudes to integral equations.

Solution of Schrödinger Equation Involving Time

The general solution of the Schrödinger equation involving the time-dependent perturbation is presented in a compact and manageable form both for periodic and aperiodic perturbations. The method includes as a special case the solution of the Schrödinger equation involving the time-independent perturbation. Formulas ready for practical uses are explicitly described. The essential part of the procedure is the determinantal method in the Laplace-transformed space. The solution is applicable to strong perturbations as well as weak perturbations.

Modified determinantal method

An improved version of the determinantal method for solving N D equations is developed. Unlike the determinantal approximation, the accuracy of this method is not limited to small coupling constants. Further, this method guarantees a real residue at the position of a pole in the partial wave amplitude whereas the determinantal approximation yields a complex residue when the pole lies on the left-hand cut in the s-plane. Both nonrelativistic and relativistic cases are considered. Numerical accuracy of the new method is examined in some typical applications.

A Proof of Nakanishi's Inequality

Nakanishi's conjectured inequality for the coefficients of the external masses and invariants in the Feynman denominators for scattering processes is proved by a determinantal method suggested by the analogy between Feynman diagrams and electrical networks.

A reciprocal bootstrap mechanism for quarks

We have considered theP-wave quark-pseudoscalar meson scattering and calculated the appropriate crossing matrix, which shows a reciprocal bootstrap relationship between quarks, Q, and some other particles, Q*, having the baryon number 1/3, spin 3/2 and belonging to the 15-dimensional representation ofSU3. Using the determinantal method we have obtained self-consistent values which areMQ⋍2429 MeV,M*Q⋍5251 MeV,g12⋍22,g22⋍32, whereMQ,M*Q andg12,g22 are respectively the masses and couplings of Q and Q*.

Bound on Screening Corrections in Beta Decay

A determinantal method is employed to provide a rigorous bound on the screening corrections in beta decay in the nonrelativistic approximation. Our results suffice to demonstrate that a previous numerical calculation of these corrections is inaccurate.

Approximate Solution to theNDEquations

An approximate solution to the N/D equations which does not have the limitations of the determinantal method developed by Baker and yields a closer approximation to the actual solution for some cases is presented. The result of a numerical example is given. The advantages of both methods are discussed. (L.B.S.)

Determinantal Method in Perturbation Theory

The determinantal method of deriving fundamental relations of a new perturbation theory, which the author has presented recently, is demonstrated. This method is simpler than the Green's function method which has been adopted in the previous paper. The Brillouin-Wigner perturbation theory is discussed for comparison.

Jost functions and determinantal method in potential scattering

Some properties of Jost functions are reviewed for real and complex angular momentum and compared to the determinantal expansion. The exact phase shifts and Jost functions in the complexk-plane are calculated forS- andP-waves and compared to the first and second-order determinantal expansion.

Application of the Variation Method to Field Quantization

The object of this paper is the application of the variation method to a simple field theory. A representation is found in which the state functions are emphasized. Variational trial forms are chosen for these, and optimized by making the expectation value of the field Hamiltonian stationary. Only the simplest case of a neutral, spin-zero, boson field with a fourth-power self-coupling term is considered here, but it is hoped that with further elaboration this may in the future lead to a description of the multipion resonances. The variation method has the advantage of avoiding any limitation on the strength of the self-coupling. Explicit results are obtained for the vacuum, single-particle, and two-particle (scattering and bound) states, and comparison is made with the determinantal method. Finally, a criterion for the variational stability of the vacuum state is obtained.

NDMethod for Complex Partial Waves

Conditions on complex partial-wave amplitudes necessary and sufficient for the validity of the Mandelstam representation have been derived for both nonrelativistic potential scattering and relativistic two-particle scattering. These conditions have been used to obtain iterative solutions to the nonrelativistic scattering problem with a given potential (restricted to superposition of Yukawa potentials), and to the relativistic problem in the elastic unitarity approximation with a given discontinuity across the left-hand cut. In the first case, the method used reduces to the well-known determinantal method for physical partial waves, and in the second case, it is a natural generalization of the $\frac{N}{D}$ method to complex values of $l$. The Regge poles, given by the zeros of the denominator, can be studied in a perturbative expansion in terms of the strength of the potential. Several applications are pointed out.

Relativistic pion-nucleon scattering with the determinantal method

Relativistic pion-nucleon scattering with the determinantal method

Angular Momentum Expansions in Relativistic Field Theory

As a step toward more general applications of angular momentum expansions in quantum field theory the expansion of the scattering matrix is examined in the case of scattering of spin 0 by spin 1/2 particles. The matrix is represented in terms of its eigenvalues and eigenvectors, the latter being eigenstates of total angular momentum. Using eigenstates of helicity to simplify the discussion, the eigenvectors raay sometimes be obtained from the conservation laws alone. The eigenvalues are computed only to second order in a Yukawa interaction, but the results are more useful than the usual second-order matrix elements. Since angular momentuni expansions lead effectively to solutions of operator equations, the expressions derived facilitate the relativistic application of Heitler-s unitary approximation (with an exact solution of the equation relating the transition operator T and the Hermitian reaction operator K) or the determinantal method of Schwinger and Baker. (auth)

LXXV. The numerical reduction of non-singular matrix pencils

Summary The problem of reducing a non-singular matrix pencil L≡l 1 L 1+l 2 L 2, |L|≢0, to canonical form, and of finding the latent roots l 1 : l 2 and root- vectors x that satisfy Lx=0, arises in many branches of applied mathematics. The most familiar of these applications is the analysis of small vibrations of a dynamical system about an equilibrium position by means of Lagrangian frequency equations ; while other applications include the conditioned stationary values of multivariate functions, approximations to certain integral equations, orthogonal mean square regressions in multivariate statistics, linear operational equations with constant coefficients, the determination of matrix commutants, matrix interpolation, and the growth behaviour of animal populations. The stock of current reduction procedures embraces the determinantal method, the Duncan-Collar-Aitken- Hotelling method, the Morris escalator, Samuel-son's method, Horst's method, and the Cayley-Hamilton method. For one reason or another, each of these methods proves somewhat unsatisfactory—either it demands considerable cpmputing labour, or it is unsuitable for automatic computors, or it only solves certain sections of the problem, or it is restricted to special cases such as symmetric matrices or distinct latent roots. In this paper I propound a new method, which I consider an improvement on current procedures. This new method is an extension of the Cayley-Hamilton method, based upon the general theory of collineations. It is quite general, not requiring modification in special cases ; it provides the complete solution of all the latent roots and all the root-vectors and/or invariant space bases ; it is suitable for punched card analysis and automatic computors—with the exception of a simple set of divisions, ring operations alone suffice for the whole computation—and it calls for rather less computing labour than current methods.

THE DETERMINANTAL METHOD OF SOLVING NORMAL EQUATIONS

In the treatises on the method of least squares published so far in the English language the application and comparison of different methods of solving normal equations has received but slight notice.