Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Schrödinger type equations with singular coefficients and lower order terms

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

Abstract

In this paper we investigate the well-posedness of the Cauchy problem for a Schrödinger operator with singular lower order terms. We allow distributional coefficients and we approach this problem via the regularising methods at the core of the theory of very weak solutions. We prove that a very weak solution exists and it is unique modulo negligible perturbations. Very weak solutions converge to classical solutions when the equation coefficients are regular enough.

Schrödinger type equations with singular coefficients and lower order terms

Abstract

In this paper we investigate the well-posedness of the Cauchy problem for a Schrödinger operator with singular lower order terms. We allow distributional coefficients and we approach this problem via the regularising methods at the core of the theory of very weak solutions. We prove that a very weak solution exists and it is unique modulo negligible perturbations. Very weak solutions converge to classical solutions when the equation coefficients are regular enough.
Paper Structure (17 sections, 13 theorems, 159 equations)

This paper contains 17 sections, 13 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Construction of the regularised problem
  4. Regularisation of $a(t)$
  5. Regularisation of $a_j(t)$, $j=0, 1, \ldots, n$
  6. Regularisation of $b_j(x)$, $j=0,1,\ldots,n$
  7. Regularisation of the Cauchy data
  8. Pseudodifferential operators
  9. Solving the regularised problem: the case $n=1$
  10. Solving the regularised problem: the $n$-dimensional case
  11. Invertibility of $e^{\lambda(x,D)}$
  12. Computing the operator $S_{\varepsilon, \lambda}$
  13. Energy estimate for the regularised problem.
  14. Proof of Theorem \ref{['mainn']}
  15. Uniqueness of the very weak solution and consistency with the regular theory
  16. ...and 2 more sections

Key Result

Proposition 2.1

Theorems & Definitions (24)

  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Definition 4.1
  • ...and 14 more