<p>Consider a solution <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$u$</EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mi>u</mi> <mo>=</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\Delta u +Vu=0$</EquationSource> </InlineEquation> on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R}^{2}$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>V</mi> </math></EquationSource> <EquationSource Format="TEX">$V$</EquationSource> </InlineEquation> is real-valued, measurable and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">|</mo> <mi>V</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$|V|\leq 1$</EquationSource> </InlineEquation>. If <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="212" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo>≤</mo> <mo>exp</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>C</mi> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <msup> <mo>log</mo> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$|u(x)| \leq \exp (-C |x| \log ^{1/2}|x|)$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo>></mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$|x|>2$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>C</mi> </math></EquationSource> <EquationSource Format="TEX">$C$</EquationSource> </InlineEquation> is a sufficiently large absolute constant, then <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1340_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>u</mi> <mo>≡</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$u\equiv 0$</EquationSource> </InlineEquation>.</p>

The Landis conjecture on exponential decay

期刊论文
出版模式：

开放获取
发表日期：

2025年6月25日

查看全文

A. Logunov,
E. Malinnikova,
N. Nadirashvili,
F. Nazarov

摘要

Consider a solution $u$ $u$ to $Δ u + V u = 0$ $\Delta u +Vu=0$ on $R^{2}$ $\mathbb{R}^{2}$ , where $V$ $V$ is real-valued, measurable and $| V | \leq 1$ $|V|\leq 1$ . If $| u (x) | \leq exp (- C | x | {log}^{1 / 2} | x |)$ $|u(x)| \leq \exp (-C |x| \log ^{1/2}|x|)$ , $| x | > 2$ $|x|>2$ , where $C$ $C$ is a sufficiently large absolute constant, then $u \equiv 0$ $u\equiv 0$ .