Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

The non autonomous Cauchy problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i\partial_{t}u=-\partial_{xx}^{2} u+\alpha(t) \delta_{0}u$\end{document} with ut = 0 = u0 is considered in L2 (ℝ) . The regularity assumptions for α are accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem.