Kernel Of Integral Equation Research Articles

On the solution of the three-particle integral equations by the separable expansion method

Use of a separable expansion for the two-particle t-matrix reduces the problem of three particles with pair interaction to a set of one-dimensional integral equations. By the subsequent separable representation of kernels of such integral equations (based on the Bateman method) the problem of three identical particles is reduced to the solution of algebraic equations.

Magnetic-Field Dependence of the Bound State Due to thes-dExchange Interaction

On the basis of the theory developed by Y osida, Okiji and Y oshimori on the singlet ground state of a system of conduction eleCtrons and a localized spin coupled with an antiferromagnetic exchange interaction, the magnetic-field dependence of the ground state is investigated· in logarithmic accuracy by collecting the most divergent terms in the integration kernel. The correctness of a previous result for the susceptibility obtained by an iteration method is confirmed in the weak coupling limit, and it is further concluded that in this limit the bound state does not disappear at a finite value of magnetic field, but approaches the normal state asymptotically at high field. recently succeeded in deriving a closed form ot solution for the singlet bound state by collecting the most divergent terms in the kernel of the secular integral equation. Further, by the use of this method, calculations have been made of such quantities as the normalization integral of the wave function, the kinetic energy and the charge density and more conclusive results of a series of work are derived. 6 ) In a previous paper/) as a part of a series of the work, we have calculated by the iteration method the magnetic susceptibility of the singlet bound state at OaK and obtained ,uB 2 /IEI for the susceptibility, where E denotes its binding energy. It was shown that this result does hold at any stage of approximation. However, as the calculation was restricted in that paper to a weak field, it was impossible to elucidate the behavior of the localized spin in high field. In high field the situation is considered qualitatively as follows. The singlet bound state, which is stable in the absence of the field and at OaK, will decrease its binding energy as field is increased, because field makes a spin-flip process due to the exchange interaction difficult. In this paper, using the method of collecting the most divergent terms in the integration kernel,5),6) we derive an expression for the binding energy as a function of the applied field, which is valid up to a

Operator Formalism for Daughter Trajectories in the Bethe-Salpeter Equation

The existence of daughter Regge poles is demonstrated by a raising and lowering operator formalism without explicit use of $O(4)$ symmetry. The commutation relations of these operators with the kernel of the Bathe-Salpeter integral equation require the existence of the degeneracy which leads to an infinite chain of daughter Regge trajectories. It is found that the chain of daughter poles does not terminate even if the parent pole is at a positive integer. A formula is given for calculating the slopes of the Regge trajectories.

The Discontinuities of the Triangle Graph as a Function of an Internal Mass. II

We consider the discontinuities of the triangle-graph amplitude as a function of an internal mass variable. These discontinuities are important, since they form the kernel of the Aitchison-Anisovich integral equation, which is derived from the Khuri-Treiman three-body final-state-interaction dispersion relation. We evaluate the discontinuities by explicitly performing the Feynman α integrations. We also discuss their analytic continuations. Finally we consider the applicability of the Cutkosky rules to such an internal mass variable discontinuity. It is argued that these rules must be modified in two ways. One of these is straightforward, having to do with the appearance of spacelike masses. The other is more involved and is a consequence of the results of homology theory. We apply the modified Cutkosky rules to the triangle-graph discontinuities and obtain the same results as found by the direct method, so confirming the modifications which we have made.

Concept of terminal admittance and impedance matrixes extended to continuous media

The concept of terminal impedance and admittance matrixes can be generalised to cover continuous media. The matrixes take the form of kernels of singular integral equations. Some properties and applications are considered, one of which establishes a rigorous connection between continuum and resistance-network calculations.

An expansion for some second-order probability distributions and its application to noise problems

In this paper it is shown that, in general, second-order probability distributions may be expanded in a certain double series involving orthogonal polynomials associated with the corresponding first-order probability distributions. Attention is restricted to those second-order probability distributions which lead to a "diagonal" form for this expansion. When such distributions are joint probability distributions for samples taken from a pair of time series, some interesting results can be demonstrated. For example, it is shown that if one of the time series undergoes an amplitude distortion in a time-varying "instantaneous" nonlinear device, the covariance function after distortion is simply proportional to that before distortion. Some simple results concerning conditional expectations are given and an extension of a theorem, due to Doob, on stationary Markov processes is presented. The relation between the "diagonal" expansion used in this paper and the Mercer expansion of the kernel of a certain linear homogeneous integral equation, is pointed out and in conclusion explicit expansions are given for three specific examples.

Self-Energy Effects on Meson-Nucleon Scattering According to the Tamm-Dancoff Method

The lowest-order Tamm-Dancoff equations for the meson-nucleon system are derived by the method of Cini, using the wave functions defined by Dyson. The contributions of the meson and nucleon self-energies to the kernel of the momentum-space integral equation are renormalized. Their effects are absorbed into the coupling ${G}^{2}$, making it momentum and energy dependent. The effective coupling turns out to exhibit an anomalous behavior, having a pole for a spacelike momentum, to which it is difficult to attach a sensible physical interpretation.

Relation between Apparent Shapes of Monoenergetic Conversion Lines and Continuous Beta-Spectra in a Magnetic Spectrometer. II

The method worked out previously for correcting distortions which affect the apparent shape of a continuous beta-spectrum has been further extended. A general solution of the integral equation governing the relation between the true and the observed spectra is given, and is applied to the correction of the experimental spectra from a 1-mg/cm2 source of Cu64. The kernel of the distortion integral equation is determined from a study of the internal conversion lines of I131 (47 and 330 kev) using sources of various thicknesses from 0.042 to 0.840 mg/cm2. It is found that the discrepancy between the experimental Cu64 negatron and positron spectra and those predicted by the Fermi (``allowed'') theory can be completely understood as a thick source effect.

Symmetrisable functions and their expansion in terms of biorthogonal functions

The purpose of this communication is to announce certain results relative to the expansion of a symmetrisable function k ( s , t ) in terms of a complete biorthogonal system of fundamental functions, which belong to k ( s , t ) regarded as the kernel of a linear integral equation. An indication of the method by which the results have been obtained is given, but no attempt is made to supply detailed proofs. Preliminary Explanations . 1. Let k ( s , t ) be a function defined in the square a ≤ s ≤ b , a ≤ t ≤ b . If a function ϒ ( s , t ) can be found which is of positive type in the square a ≤ s ≤ b , a ≤ t ≤ b and such that ∫ a b ϒ ( s , x ) k ( x , t ) dx is a symmetric function of s and t , k ( s , t ) is said to be symmetrisable on the left by ϒ ( s , t ) is the square. Similarly, if a function ϒ' ( s, t ) of positive type can be found such that ∫ a b k ( s , x ) ϒ' ( x , t ) dx is a symmetric function of s and t , k ( s , t ) is said to be symmetrisable on the right by ϒ' ( s , t ).