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<p>For Schrödinger operators <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H_V=-\Delta _g+V\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>V</mi> </msub> <mo>=</mo> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>g</mi> </msub> <mo>+</mo> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation> with critically singular potentials <i>V</i> on compact manifolds, we prove sharp estimates for the restriction of eigenfunctions to submanifolds. Our method refines the perturbative argument by Blair et al. (J Geom Anal 31(7):6624–6661, 2021) and enables us to deal with submanifolds of all codimensions. As applications, we obtain improved estimates on negatively curved manifolds and flat tori. In particular, we extend the uniform <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> restriction estimates on flat tori by Bourgain and Rudnick (Geom Funct Anal 22(4):878–937, 2012) to singular potentials.</p>

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Restriction of Schrödinger Eigenfunctions to Submanifolds

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Xiaoqi Huang,
  • Xing Wang,
  • Cheng Zhang

摘要

For Schrödinger operators \(H_V=-\Delta _g+V\) H V = - Δ g + V with critically singular potentials V on compact manifolds, we prove sharp estimates for the restriction of eigenfunctions to submanifolds. Our method refines the perturbative argument by Blair et al. (J Geom Anal 31(7):6624–6661, 2021) and enables us to deal with submanifolds of all codimensions. As applications, we obtain improved estimates on negatively curved manifolds and flat tori. In particular, we extend the uniform \(L^2\) L 2 restriction estimates on flat tori by Bourgain and Rudnick (Geom Funct Anal 22(4):878–937, 2012) to singular potentials.