We characterize the domain of the Schrödinger operators <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2024_1021_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="126" /> </InlineMediaObject> <EquationSource Format="TEX">$S=-\Delta +c|x|^{-\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <msup> <mrow> <mi>c</mi> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2024_1021_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p(\mathbb {R}^N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2024_1021_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\alpha <2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2024_1021_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$c\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2024_1021_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha p< N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mi>p</mi> <mo><</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, the domain characterization is essentially known and can be proved using different tools, for instance, kernel estimates and potentials in the Kato class or in the reverse Hölder class. However, the other cases seem not to be known so far. In this paper, we give the explicit description of the domain of S for all ranges of parameters <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2024_1021_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$p,\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>, and c.

Domain characterization for Schrödinger operators with sub-quadratic singularity

期刊论文
发表日期：

2025年1月15日

查看全文

Giorgio Metafune,
Motohiro Sobajima

摘要

We characterize the domain of the Schrödinger operators $S=-\Delta +c|x|^{-\alpha }$ $S = - Δ + {c | x |}^{- α}$ in $L^p(\mathbb {R}^N)$ $L^{p} (R^{N})$ , with $0<\alpha <2$ $0 < α < 2$ and $c\in \mathbb {R}$ $c \in R$ . When $\alpha p< N$ $α p < N$ , the domain characterization is essentially known and can be proved using different tools, for instance, kernel estimates and potentials in the Kato class or in the reverse Hölder class. However, the other cases seem not to be known so far. In this paper, we give the explicit description of the domain of S for all ranges of parameters $p,\alpha $ $p, α$ , and c.