In this article, we investigate the behavior of solutions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ u(x,t) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to the fractional Schrödinger equation on rank one symmetric spaces of non-compact type. We proved that as time <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ t $</EquationSource> <EquationSource Format="MATHML"><math> <mi>t</mi> </math></EquationSource> </InlineEquation> approaches 0, then u(x, t) converges pointwise almost everywhere to the initial radial data <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ f $</EquationSource> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> </InlineEquation>, provided that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ f \in H^s(\mathbb {X}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ s > \frac{1}{2} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. This result extends Sjölin’s results in this setting.

Regularity Results for the Schrödinger Equation on Rank One Symmetric Spaces of Non-Compact Type

期刊论文
发表日期：

2026年2月10日

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Pratyoosh Kumar,
Manali Sajjan

摘要

In this article, we investigate the behavior of solutions $ u(x,t) $ $u (x, t)$ to the fractional Schrödinger equation on rank one symmetric spaces of non-compact type. We proved that as time $ t $ $t$ approaches 0, then u(x, t) converges pointwise almost everywhere to the initial radial data $ f $ $f$ , provided that $ f \in H^s(\mathbb {X}) $ $f \in H^{s} (X)$ with $ s > \frac{1}{2} $ $s > \frac{1}{2}$ . This result extends Sjölin’s results in this setting.