A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients

We construct a generalization of the multiplicative product of distributions presented by L. H\"ormander in [L. H\"ormander, {\it The analysis of linear partial differential operators I} (Springer-Verlag, 1983)]. The new product is defined in the vector space {\mathcal A}(\bkR) of piecewise smooth functions f: \bkR \to \bkC and all their (distributional) derivatives. It is associative, satisfies the Leibniz rule and reproduces the usual pointwise product of functions for regular distributions in {\mathcal A}(\bkR). Endowed with this product, the space {\mathcal A}(\bkR) becomes a differential associative algebra of generalized functions. By working in the new {\mathcal A}(\bkR)-setting we determine a method for transforming an ordinary linear differential equation with general solution $\psi$ into another, ordinary linear differential equation, with general solution $\chi_{\Omega} \psi$, where $\chi_{\Omega}$ is the characteristic function of some prescribed interval \Omega \subset \bkR.Comment: 23 pages, Latex fil

On the Geroch-Traschen class of metrics

We compare two approaches to semi-Riemannian metrics of low regularity. The maximally 'reasonable' distributional setting of Geroch and Traschen is shown to be consistently contained in the more general setting of nonlinear distributional geometry in the sense of Colombea

Distributional approach to point interactions in one-dimensional quantum mechanics

We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schr\"odinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called $\delta^\prime$ interaction.Comment: Open Access link: http://journal.frontiersin.org/Journal/10.3389/fphy.2014.00023/abstrac

Tensor Distributions in the Presence of Degenerate Metrics

Tensor distributions and their derivatives are described without assuming the presence of a metric. This provides a natural framework for discussing tensor distributions on manifolds with degenerate metrics, including in particular metrics which change signature.Comment: REVTeX, 19 pages; submitted to IJMP