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<p>In this paper, we give an alternative perspective of the criticality theory for (nonnegative) Schrödinger operators. Schrödinger operator <i>S</i> is classified as subcritical/critical in terms of the existence/nonexistence of a positive Green function for the associated elliptic equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Su=f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation>. Such a property strongly affects to the large-time behavior of solutions to the parabolic equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\partial _tv+Sv=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>v</mi> <mo>+</mo> <mi>S</mi> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we propose a remarkable quantity in terms of the structure of Hilbert lattices, which keeps some important properties including the notion of criticality theory. As an application, we study the large-time behavior of solutions to the hyperbolic equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\partial _t^2w+Sw=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>∂</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mi>w</mi> <mo>+</mo> <mi>S</mi> <mi>w</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Remarks on criticality theory for Schrödinger operators and its application to wave equations with potentials

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  • 发表日期：

    2026年1月1日

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  • Motohiro Sobajima

摘要

In this paper, we give an alternative perspective of the criticality theory for (nonnegative) Schrödinger operators. Schrödinger operator S is classified as subcritical/critical in terms of the existence/nonexistence of a positive Green function for the associated elliptic equation \(Su=f\) S u = f . Such a property strongly affects to the large-time behavior of solutions to the parabolic equation \(\partial _tv+Sv=0\) ∂ t v + S v = 0 . In this paper, we propose a remarkable quantity in terms of the structure of Hilbert lattices, which keeps some important properties including the notion of criticality theory. As an application, we study the large-time behavior of solutions to the hyperbolic equation \(\partial _t^2w+Sw=0\) ∂ t 2 w + S w = 0 .