Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Perturbed Fourier Transform Associated with Schrödinger Operators

Shijun Zheng

TL;DR

This work develops an $L^2$-theory for the perturbed Fourier transform $\mathcal{F}$ associated with a one-dimensional Schrödinger operator $H=H_0+V$, where $V$ is a real-valued finite measure or belongs to $L^1\cap L^2$. It constructs generalized eigenfunctions $e(x,\xi)$ via the Lippmann–Schwinger framework and proves a Plancherel-type identity $\mathcal{F}^*\mathcal{F}=P_{ac}$ with surjectivity of $\mathcal{F}$ onto the absolutely continuous subspace, yielding an explicit spectral calculus and kernel formulas for spectral operators. The analysis leverages Green’s functions, Jost solutions, and resolvent techniques to connect the perturbed and free evolutions through wave operators $W_\pm$, establishing asymptotic completeness and detailed scattering behavior for $e^{-itH}$. The results unify time-dependent scattering with a concrete $L^2$-theoretic eigenfunction expansion, providing explicit integral kernels and a robust framework for spectral and scattering analysis of measure-valued and short-range potentials.

Abstract

We give an exposition on the $L^2$ theory of the perturbed Fourier transform associated with a Schrödinger operator $H=-d^2/dx^2 +V$ on the real line, where $V$ is a real-valued \mbox{finite} measure. In the case $V\in L^1\cap L^2$, we explicitly define the perturbed Fourier transform $\mathcal{F}$ for $H$ and obtain an eigenfunction expansion theorem for square integrable functions. This provides a complete proof of the inversion formula for $\cF$ that covers the class of short range potentials in $(1+|x|)^{-\frac12-\eps} L^2 $. Such paradigm has applications in the study of scattering problems in connection with the spectral properties and asymptotic completeness of the wave operators.

Perturbed Fourier Transform Associated with Schrödinger Operators

TL;DR

This work develops an -theory for the perturbed Fourier transform associated with a one-dimensional Schrödinger operator , where is a real-valued finite measure or belongs to . It constructs generalized eigenfunctions via the Lippmann–Schwinger framework and proves a Plancherel-type identity with surjectivity of onto the absolutely continuous subspace, yielding an explicit spectral calculus and kernel formulas for spectral operators. The analysis leverages Green’s functions, Jost solutions, and resolvent techniques to connect the perturbed and free evolutions through wave operators , establishing asymptotic completeness and detailed scattering behavior for . The results unify time-dependent scattering with a concrete -theoretic eigenfunction expansion, providing explicit integral kernels and a robust framework for spectral and scattering analysis of measure-valued and short-range potentials.

Abstract

We give an exposition on the theory of the perturbed Fourier transform associated with a Schrödinger operator on the real line, where is a real-valued \mbox{finite} measure. In the case , we explicitly define the perturbed Fourier transform for and obtain an eigenfunction expansion theorem for square integrable functions. This provides a complete proof of the inversion formula for that covers the class of short range potentials in . Such paradigm has applications in the study of scattering problems in connection with the spectral properties and asymptotic completeness of the wave operators.

Paper Structure

This paper contains 15 sections, 24 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the Schrödinger operator
  3. Finite measures and their topology
  4. The hamiltonian $H:=H_V$ defined as quadratic form
  5. Green's function and Jost functions
  6. Scattering solutions
  7. Jost functions and scattering solutions
  8. Perturbed Fourier transforms $\operatorname{\mathscr{F}}$ and $\operatorname{\mathscr{F}}^*$
  9. The $L^2$ theory for $H_V$, $V\in L^1\cap L^2$
  10. The resolvent for $H_V$
  11. Proof of Theorem \ref{['th:Plancherel']}
  12. Wave operators and surjectivity of ${\mathcal{F}}$
  13. Wave operator $\Omega$ and ${\mathcal{F}}$
  14. Integral representation for spectral operators
  15. Wave operators and scattering of $e^{-itH}$

Key Result

Proposition 3.1

Let $V$ satisfy the condition (eV:wei2). Let $\Im z\ge 0$ and $\alpha_z\ne 0$. Then $\varphi_z^{\pm}$ have the following properties.

Theorems & Definitions (62)

  • Proposition 3.1
  • Remark 1
  • Remark 2
  • Proposition 3.2: Spectral measure for $H_V$
  • Example 1: Pöschl-Teller potential OZh06
  • Example 2: $\delta$ potential
  • Proposition 4.1
  • Theorem 4.2
  • Theorem 5.1
  • Remark 3
  • ...and 52 more