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<p>We investigate the negative part of the spectrum of the operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(-\partial ^2 - \mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msup> <mi>∂</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where a locally finite Radon measure <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mu \ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> serves as a potential. We obtain estimates for the eigenvalue counting function, for individual eigenvalues and estimates of the Lieb–Thirring type. A crucial tool for our estimates is Otelbaev’s function, a certain average of the measure-potential <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, which is used both in the proofs and the formulation of most of the results.</p>

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Negative Eigenvalue Estimates for the 1D Schrödinger Operator with Measure-Potential

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  • 发表日期：

    2025年2月17日

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  • Robert Fulsche,
  • Medet Nursultanov,
  • Grigori Rozenblum

摘要

We investigate the negative part of the spectrum of the operator \(-\partial ^2 - \mu \) - ∂ 2 - μ on \(L^2(\mathbb {R})\) L 2 ( R ) , where a locally finite Radon measure \(\mu \ge 0\) μ ≥ 0 serves as a potential. We obtain estimates for the eigenvalue counting function, for individual eigenvalues and estimates of the Lieb–Thirring type. A crucial tool for our estimates is Otelbaev’s function, a certain average of the measure-potential \(\mu \) μ , which is used both in the proofs and the formulation of most of the results.