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<p>We consider dissipative Schrödinger operators of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H=-\Delta +V(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>=</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textsf{L}^{\,\!\!2}(\mathbb {R}^3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="sans-serif">L</mi> <mrow> <mspace width="0.166667em" /> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with <i>V</i>(<i>x</i>) a complex, bounded and decaying potential with a non-positive imaginary part. We prove a topological version of Levinson’s theorem corresponding to an index theorem for the discrete, complex spectrum of <i>H</i>.</p>

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Levinson’s Theorem for Dissipative Systems, or How to Count the Asymptotically Disappearing States

  • 期刊论文

  • 发表日期：

    2026年3月6日

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  • A. Alexander,
  • J. Faupin,
  • S. Richard

摘要

We consider dissipative Schrödinger operators of the form \(H=-\Delta +V(x)\) H = - Δ + V ( x ) on \(\textsf{L}^{\,\!\!2}(\mathbb {R}^3)\) L 2 ( R 3 ) , with V(x) a complex, bounded and decaying potential with a non-positive imaginary part. We prove a topological version of Levinson’s theorem corresponding to an index theorem for the discrete, complex spectrum of H.