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Abstract <p> For a model nonstationary Schrödinger equation with a potential depending on time <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation> and with a small parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation> in front of the derivative with respect to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation>, we investigate a solution that in the case of a potential independent of time has the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(e^{-iE\tau/\varepsilon} \psi(x,E)\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\psi(\cdot,E)\)</EquationSource> </InlineEquation> is a generalized eigenfunction of the stationary Schrödinger operator corresponding to a value <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(E\)</EquationSource> </InlineEquation> of the spectral parameter, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(E\)</EquationSource> </InlineEquation> being in the continuous spectrum. </p>

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Adiabatic Evolution Generated by a Schrödinger Operator with a Continuous Spectrum

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • A. A. Fedotov,
  • V. A. Sergeev

摘要

Abstract

For a model nonstationary Schrödinger equation with a potential depending on time \(\tau\) and with a small parameter \(\varepsilon\) in front of the derivative with respect to \(\tau\) , we investigate a solution that in the case of a potential independent of time has the form \(e^{-iE\tau/\varepsilon} \psi(x,E)\) , where \(\psi(\cdot,E)\) is a generalized eigenfunction of the stationary Schrödinger operator corresponding to a value \(E\) of the spectral parameter, \(E\) being in the continuous spectrum.