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Schwartz very weak solutions for Schrödinger type equations with distributional coefficients

Alexandre Arias Junior, Alessia Ascanelli, Marco Cappiello, Claudia Garetto

TL;DR

This work extends well-posedness theory for Schrödinger-type equations to coefficients that are tempered distributions in space and continuous in time by introducing Schwartz very weak solutions. It regularises coefficients and data with Schwartz mollifiers and embeds the problem into the Colombeau algebra $\mathcal{G}_{\mathscr S}$, proving existence and uniqueness of a Schwartz $\mathscr{S}$-very weak solution modulo $\mathscr{S}$-negligible perturbations. A central technical contribution is a two-step pseudodifferential conjugation, using $e^{\lambda_{1}}(x,D)$ and $e^{\lambda_{2}}(x,D)$, which yields robust energy estimates and controls the decay at infinity, enabling passage from the regularised to the original problem and establishing consistency with classical Schwartz theory when coefficients are regular. The results thus provide a principled framework for well-posedness of Schrödinger-type problems with highly irregular coefficients, with clear connections to classical results and a rigorous energy-estimate machinery.

Abstract

This paper continues the analysis of Schrödinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are continuous in time. We prove that the corresponding Cauchy problem, which in general cannot even be stated in the standard distributional setting, admits a Schwartz very weak solution which is unique modulo negligible perturbations. Consistency with the classical theory is proved in the case of regular coefficients and Schwartz Cauchy data.

Schwartz very weak solutions for Schrödinger type equations with distributional coefficients

TL;DR

This work extends well-posedness theory for Schrödinger-type equations to coefficients that are tempered distributions in space and continuous in time by introducing Schwartz very weak solutions. It regularises coefficients and data with Schwartz mollifiers and embeds the problem into the Colombeau algebra , proving existence and uniqueness of a Schwartz -very weak solution modulo -negligible perturbations. A central technical contribution is a two-step pseudodifferential conjugation, using and , which yields robust energy estimates and controls the decay at infinity, enabling passage from the regularised to the original problem and establishing consistency with classical Schwartz theory when coefficients are regular. The results thus provide a principled framework for well-posedness of Schrödinger-type problems with highly irregular coefficients, with clear connections to classical results and a rigorous energy-estimate machinery.

Abstract

This paper continues the analysis of Schrödinger type equations with distributional coefficients initiated by the authors in [3]. Here we consider coefficients that are tempered distributions with respect to the space variable and are continuous in time. We prove that the corresponding Cauchy problem, which in general cannot even be stated in the standard distributional setting, admits a Schwartz very weak solution which is unique modulo negligible perturbations. Consistency with the classical theory is proved in the case of regular coefficients and Schwartz Cauchy data.
Paper Structure (18 sections, 18 theorems, 166 equations)

This paper contains 18 sections, 18 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Schwartz very weak solutions and main result
  3. Regularisation of $c_j(t,x)$, $j=0,1,\ldots,n$
  4. Regularisation of the Cauchy data
  5. Solving the regularised problem \ref{['regularised_CP_true']}
  6. From the regularised problem to the original problem
  7. Existence
  8. Uniqueness
  9. Proof of Proposition \ref{['pompem']}
  10. Invertibility of $e^{\lambda_1}(x,D)$ and conjugation theorem
  11. Invertibility of $e^{\lambda_2}(x,D)$
  12. Conjugation of $S_{\varepsilon,s}$ by $e^{\lambda_1}(x,D)$
  13. Conjugation of $S_{\varepsilon, s, \lambda_1}$ by $e^{\lambda_2}(x,D)$
  14. Choices of $M_1$, $M_2$, $h$ and $L^2$ energy estimate
  15. Consistency with the classical theory in the case of regular coefficients
  16. ...and 3 more sections

Key Result

Proposition 2.1

Theorems & Definitions (35)

  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • Remark 2.8
  • Proposition 3.1
  • Theorem 3.2
  • ...and 25 more