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Abstract <p> We consider the Schrödinger equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\(ih\partial_t\psi=H\psi\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="139" /> </InlineMediaObject> <EquationSource Format="TEX">\(\psi=\psi(\cdot,t)\in L^2(\mathbb{T})\)</EquationSource> </InlineEquation>. The operator <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">\(H=-\partial^2_x+V(x,t)\)</EquationSource> </InlineEquation> includes a smooth potential <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(V\)</EquationSource> </InlineEquation>, which is assumed to be time <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(T\)</EquationSource> </InlineEquation>-periodic. Let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(W=W(t)\)</EquationSource> </InlineEquation> be the fundamental solution of this linear ODE system on <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2(\mathbb{T})\)</EquationSource> </InlineEquation>. Then, according to the terminology from Lyapunov–Floquet theory, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal M=W(T)\)</EquationSource> </InlineEquation> is the monodromy operator. We prove that <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal M\)</EquationSource> </InlineEquation> is unitarily conjugated to <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq10.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(D+\mathcal C\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(D\)</EquationSource> </InlineEquation> is diagonal in the standard Fourier basis, while <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10688_2025_1168_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal C\)</EquationSource> </InlineEquation> is a compact operator with an arbitrarily small norm. </p>

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On Quantum Floquet Theorem

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • Dmitry Treschev

摘要

Abstract

We consider the Schrödinger equation \(ih\partial_t\psi=H\psi\) , \(\psi=\psi(\cdot,t)\in L^2(\mathbb{T})\) . The operator \(H=-\partial^2_x+V(x,t)\) includes a smooth potential \(V\) , which is assumed to be time \(T\) -periodic. Let \(W=W(t)\) be the fundamental solution of this linear ODE system on \(L^2(\mathbb{T})\) . Then, according to the terminology from Lyapunov–Floquet theory, \(\mathcal M=W(T)\) is the monodromy operator. We prove that \(\mathcal M\) is unitarily conjugated to \(D+\mathcal C\) , where \(D\) is diagonal in the standard Fourier basis, while \(\mathcal C\) is a compact operator with an arbitrarily small norm.