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<p>Weyl <i>m</i>-function characterisations can be used to describe the behaviour of eigenvalues of Schrödinger operators with point interactions. Using complex scaling, transformators and a local Borg-Marchenko theorem we describe the asymptotic behaviour of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(m(\lambda )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(|\lambda |\rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>λ</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> in different regions of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>. Our asymptotic results are applied to the evolution of real eigenvalues of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\delta '\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>δ</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> perturbations of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\mathcal{P}\mathcal{T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation>-symmetric problems as functions of coupling constants.</p>

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Weyl-function asymptotics with applications to \(\mathcal{P}\mathcal{T}\)-symmetry and point interactions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月25日

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  • Marco Marletta,
  • Iveta Semorádová

摘要

Weyl m-function characterisations can be used to describe the behaviour of eigenvalues of Schrödinger operators with point interactions. Using complex scaling, transformators and a local Borg-Marchenko theorem we describe the asymptotic behaviour of \(m(\lambda )\) m ( λ ) as \(|\lambda |\rightarrow \infty \) | λ | → ∞ in different regions of \(\mathbb {C}\) C . Our asymptotic results are applied to the evolution of real eigenvalues of \(\delta \) δ and \(\delta '\) δ ′ perturbations of \(\mathcal{P}\mathcal{T}\) P T -symmetric problems as functions of coupling constants.