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On the definition of zero resonances for the Schr{ö}dinger operator with optimal scaling potentials

Viviana Grasselli

Abstract

We consider the Schr{ö}dinger operator --$Δ$ + V on the Euclidean space with potential in the Lorentz space L^{n/2,1} and we find necessary and sufficient conditions for zero to be a resonance or an eigenvalue. We consider functions with gradient in L^2 and that verify the equation (--$Δ$ + V)$ψ$ = 0, namely the kernel of (--$Δ$ + V) in the homogeneous Sobolev space of order one. We prove that a function in this set is either in a weak Lebesgue space or in L^2 , in the latter case we have a zero eigenfunction. The set of eigenfunctions is the hyperplane of functions that are orthogonal to V, furthermore we show that under some classic orthogonality conditions a zero eigenfunction belongs to the weak Lebesgue space of order one or to L^1. We study dimensions n $\ge$ 3 and in dimension three we generalize a result proved by Beceanu.

On the definition of zero resonances for the Schr{ö}dinger operator with optimal scaling potentials

Abstract

We consider the Schr{ö}dinger operator -- + V on the Euclidean space with potential in the Lorentz space L^{n/2,1} and we find necessary and sufficient conditions for zero to be a resonance or an eigenvalue. We consider functions with gradient in L^2 and that verify the equation (-- + V) = 0, namely the kernel of (-- + V) in the homogeneous Sobolev space of order one. We prove that a function in this set is either in a weak Lebesgue space or in L^2 , in the latter case we have a zero eigenfunction. The set of eigenfunctions is the hyperplane of functions that are orthogonal to V, furthermore we show that under some classic orthogonality conditions a zero eigenfunction belongs to the weak Lebesgue space of order one or to L^1. We study dimensions n 3 and in dimension three we generalize a result proved by Beceanu.
Paper Structure (5 sections, 9 theorems, 116 equations)

This paper contains 5 sections, 9 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Green function for a small potential
  3. Properties of a zero resonant state
  4. Facts about Lorentz spaces
  5. Proof of Lemma \ref{['lemma: algebraic ineq']}.

Key Result

Theorem 1.1

Let $n\geq 3$, $V \in L^{n/2,1}(\mathbb{R}^n)$ and $\psi \in \dot{H}^1(\mathbb{R}^n)$ a solution of the equation $(-\Delta + V)\psi =0$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Remark 2
  • Lemma 2.1
  • proof
  • Remark 3
  • Lemma 2.2
  • proof
  • Remark 4
  • ...and 14 more