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Characterization of infinitesimal boundedness of Schrödinger operator

Yanhan Chen

TL;DR

The paper characterizes weighted infinitesimal boundedness for the fractional Schrödinger operator $H=(-\Delta)^{\alpha/2}+V$ in $L^{p}(w)$, proving equivalence between an $\varepsilon$-type bound and Carleson-type capacity conditions expressed via weighted Riesz/Bessel capacities. It develops a sparse domination framework to connect resolvent estimates with these capacitary conditions and extends Maz'ya–Verbitsky’s classical results to the weighted setting, including localization and capacity-theoretic methods. The work also provides corollaries for power weights and higher-order Trudinger-type inequalities, supported by weighted Poincaré and Gagliardo–Nirenberg inequalities, with implications for self-adjointness and spectral stability of Schrödinger operators. Overall, the results offer a robust capacitary characterization of infinitesimal boundedness in weighted fractional settings, along with practical criteria and corollaries for specific weight/potential choices.

Abstract

In this paper, we characterize the weighted infinitesimal boundedness: for $0<α<n$ and $1<p<\infty$, $$\|Vφ\|_{L^{p}(w)}^{p}\leqε\|(-Δ)^{\fracα{2}}φ\|_{L^{p}(w)}^{p}+C(ε)\|φ\|_{L^{p}(w)}^{p}.$$ In particular, we extend the classical result due to Maz'ya and Verbitsky by using Carleson condition, localization estimates and capacity theory.

Characterization of infinitesimal boundedness of Schrödinger operator

TL;DR

The paper characterizes weighted infinitesimal boundedness for the fractional Schrödinger operator in , proving equivalence between an -type bound and Carleson-type capacity conditions expressed via weighted Riesz/Bessel capacities. It develops a sparse domination framework to connect resolvent estimates with these capacitary conditions and extends Maz'ya–Verbitsky’s classical results to the weighted setting, including localization and capacity-theoretic methods. The work also provides corollaries for power weights and higher-order Trudinger-type inequalities, supported by weighted Poincaré and Gagliardo–Nirenberg inequalities, with implications for self-adjointness and spectral stability of Schrödinger operators. Overall, the results offer a robust capacitary characterization of infinitesimal boundedness in weighted fractional settings, along with practical criteria and corollaries for specific weight/potential choices.

Abstract

In this paper, we characterize the weighted infinitesimal boundedness: for and , In particular, we extend the classical result due to Maz'ya and Verbitsky by using Carleson condition, localization estimates and capacity theory.

Paper Structure

This paper contains 12 sections, 16 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Weight Class
  4. Bessel Functions
  5. Sparse Families
  6. Proof of Theorem 1.1 and Corollary 1.3
  7. Equivalence between $\textnormal{(i)}$ and $\textnormal{(ii)}$ in Theorem 1.1
  8. Proof of Corollary 1.3
  9. Equivalence between $\textnormal{(ii)}$ and $\textnormal{(iii)}$ in Theorem 1.1
  10. Proof of Theorem 1.4, 1.5 and Corollary 1.6
  11. Proof of Theorem 1.4
  12. Proof of Theorem 1.5 and Corollary 1.6

Key Result

Theorem 1.1

Let $1<p<\infty$, $0<\alpha<n$, $w\in A_{p}$ and $0\leq V\in L^{p}_{loc}\cap L^{p}_{loc}(w)\cap L^{p}_{loc}(\sigma)$ with $\sigma:=w^{1-p^{\prime}}$. For $\textnormal{(i)}\sim\textnormal{(ii)}$ below, we have $\textnormal{(i)}\leftrightarrow\textnormal{(ii)}$ and $\textnormal{(iii)}\rightarrow\textn (ii) With $d\mu w:=|V|^{p}wdx$, where the first supremum is taken oven all sparse collections $\ma

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Lemma 2.2
  • ...and 24 more