Ordinary differential equations with point interactions: An inverse problem

Given a linear ordinary differential equation (ODE) on $\text{RE}$ and a set of interface conditions at a finite set of points $I\subset \text{RE}$, we consider the problem of determining another differential equation whose {\it global} solutions satisfy the original ODE on $\text{RE}\mathrm{\backslash }I$, and the interface conditions at $I$. Using an extension of the product of distributions with non-intersecting singular supports presented in [L. H"ormander, The Analysis of Linear Partial Diffe-rential Operators I, Springer-Verlag, 1983], we determine an {\it intrinsic} solution of this problem, i.e. a new ODE, satisfying the required conditions, and strictly defined within the space of Schwartz distributions. Using the same formalism, we determine a singular perturbation formulation for the $n$-th order derivative operator with interface conditions.