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<p>In this paper, we consider the Schrödinger operator on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2(0,\infty )\)</EquationSource> </InlineEquation> given by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((Hu)(x)=-u''(x)+V(x)u(x)\)</EquationSource> </InlineEquation> with a self-adjoint boundary condition at 0, where <i>V</i>(<i>x</i>) is the real perturbation. It is well known that under <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2(0,\infty )\)</EquationSource> </InlineEquation> perturbations the absolutely continuous spectrum of <i>H</i> on the positive semi-axis is preserved. In this paper, by the technique of modified Prüfer transformation and constructive methods, we prove that, with a class of smooth perturbations, <i>H</i> has exactly the given eigenvalues embedded into the absolutely continuous spectrum.</p>

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On the Schrödinger Operators with Eigenvalues Embedded into Absolutely Continuous Spectrum

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  • 发表日期：

    2026年4月3日

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  • Kang Lyu,
  • Chuanfu Yang

摘要

In this paper, we consider the Schrödinger operator on \(L^2(0,\infty )\) given by \((Hu)(x)=-u''(x)+V(x)u(x)\) with a self-adjoint boundary condition at 0, where V(x) is the real perturbation. It is well known that under \(L^2(0,\infty )\) perturbations the absolutely continuous spectrum of H on the positive semi-axis is preserved. In this paper, by the technique of modified Prüfer transformation and constructive methods, we prove that, with a class of smooth perturbations, H has exactly the given eigenvalues embedded into the absolutely continuous spectrum.