A guided tour of new tempered distributions

Laurent Schwartz, the principle architect of distribution theory, presented the impossibility of extending a form of multiplication to distribution theory. There have been many varieties of partial solutions to this problem. Some of the solutions contain heuristic computations done by physicists in quantum field theory. A recent strategy developed by J. Colombeau culminates with multiplication and integration theory for distributions. This paper develops this theory in the spirit of a sequence approach, much like fundamental sequences are to distributions. However, in the new tempered distribution theory the sequences can be noncountable. T. Todorov developed these techniques for new distributions. However, since so many applications require Fourier analysis, the new tempered distributions provide a natural setting for physics and signal analysis. The paper illustrates the product of two Dirac delta functionals,(x)(x). Other nonregular distributional products can also be computed in the same manner. The paper culminates with a new application of annihilation and creation operators in quantum field theory.     

A New Spherical Bessel Function Result Related to Quantum Mechanical Scattering Theory

The authors present a derivative formula for the square of a spherical Bessel function in terms of the spherical Bessel function of twice the argument. This derivative formula is then applied in an inversion problem for the partial-wave Born approximation in quantum mechanical scattering theory. Several other closely related results and derivative formulas are also considered.