Abstract. Using an extension of the Hörmander product of distributions, we obtain an intrinsic formulation of one-dimensional Schrödinger operators with singular potentials. This formulation is entirely defined in terms of standard Schwartz distributions and does not require (as some previous approaches) the use of more general distributions or generalized functions. We determine, in the new formulation, the action and domain of the Schrödinger operators with arbitrary singular boundary potentials. We also consider the inverse problem, and obtain a general procedure for constructing the singular (pseudo) potential that imposes a specific set of (local) boundary conditions. This procedure is used to determine the boundary operators for the complete four-parameter family of one-dimensional Schrödinger operators with a point interaction. Finally, the δ and δ ′ potentials are studied in detail, and the corresponding Schrödinger operators are shown to coincide with the norm resolvent limit of specific sequences of Schrödinger operators with regular potentials.
We study a class of linear ordinary differential equations (ODEs) with distributional coefficients. These equations are defined using an intrinsic multiplicative product of Schwartz distributions which is an extension of the Hörmander product of distributions with nonintersecting singular supports (Hörmander in The analysis of linear partial differential operators I, Springer, Berlin, 1983). We provide a regularization procedure for these ODEs and prove an existence and uniqueness theorem for their solutions. We also determine the conditions for which the solutions are regular and distributional. These results are used to study the Euler-Bernoulli beam equation with discontinuous and singular coefficients. This problem was addressed in the past using intrinsic products (under some restrictive conditions) and the Colombeau formalism (in the general case). Here we present a new intrinsic formulation that is simpler and more general. As an application, the case of a non-uniform static beam displaying structural cracks is discussed in some detail.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.