We study Schrödinger operators on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10551_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {R}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation><Equation ID="Equ49"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10551_Article_Equ49.gif" Format="GIF" Height="48" Rendition="HTML" Resolution="72" Type="Linedraw" Width="268" /> </MediaObject> <EquationSource Format="TEX">$ H = \left( -\frac{\partial ^2}{\partial x_1^2}\right) ^{\alpha /2} + \left( -\frac{\partial ^2}{\partial x_2^2}\right) ^{\alpha /2} + V, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>H</mi> <mo>=</mo> <msup> <mfenced close=")" open="("> <mo>-</mo> <mfrac> <msup> <mi>∂</mi> <mn>2</mn> </msup> <mrow> <mi>∂</mi> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> </mfenced> <mrow> <mi>α</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mfenced close=")" open="("> <mo>-</mo> <mfrac> <msup> <mi>∂</mi> <mn>2</mn> </msup> <mrow> <mi>∂</mi> <msubsup> <mi>x</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> </mfenced> <mrow> <mi>α</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>V</mi> <mo>,</mo> </mrow> </math></EquationSource> </Equation>for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10551_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in (0,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and some sufficiently regular, radial, confining potentials V. We obtain necessary and sufficient conditions on intrinsic ultracontractivity for semigroups <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10551_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">$\{e^{-tH}: \, t \ge 0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>t</mi> <mi>H</mi> </mrow> </msup> <mo>:</mo> <mspace width="0.166667em" /> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. We also get sharp estimates of first eigenfunctions of H.

Intrinsic ultracontractivity for Schrödinger semigroups based on cylindrical fractional Laplacian on the plane

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2025年7月7日

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Tadeusz Kulczycki,
Kinga Sztonyk

摘要

We study Schrödinger operators on ${\mathbb {R}}^2$ $R^{2}$ $ H = \left( -\frac{\partial ^2}{\partial x_1^2}\right) ^{\alpha /2} + \left( -\frac{\partial ^2}{\partial x_2^2}\right) ^{\alpha /2} + V, $ $H = {(- \frac{\partial^{2}}{\partial x_{1}^{2}})}^{α / 2} + {(- \frac{\partial^{2}}{\partial x_{2}^{2}})}^{α / 2} + V,$ for $\alpha \in (0,2)$ $α \in (0, 2)$ and some sufficiently regular, radial, confining potentials V. We obtain necessary and sufficient conditions on intrinsic ultracontractivity for semigroups $\{e^{-tH}: \, t \ge 0\}$ ${e^{- t H} : t \geq 0}$ . We also get sharp estimates of first eigenfunctions of H.