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<p>In this paper, we give the definition of Maslov-type index of the discrete Hamiltonian system, and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of the case <i>ω</i> = 1 to <i>ω</i> ∈ <b>U</b> degenerate case via direct method which is different from that of the known literatures. Moreover the well-posedness of the splitting numbers <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_2580_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\cal{S}_{h,\omega}^{\pm}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mrow> <mi mathvariant="script">h</mi> <mo class="MJX-tex-caligraphic" mathvariant="script">,</mo> <mi>ω</mi> </mrow> <mrow> <mo>±</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> is proven, then the index iteration theories of Bott and Long are also valid for the discrete case, and those can be also applied to the study of the symplectic algorithm.</p>

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Morse Index and Maslov-type Index of the Discrete Hamiltonian System

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Gaosheng Zhu

摘要

In this paper, we give the definition of Maslov-type index of the discrete Hamiltonian system, and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of the case ω = 1 to ω ∈ U degenerate case via direct method which is different from that of the known literatures. Moreover the well-posedness of the splitting numbers \(\cal{S}_{h,\omega}^{\pm}\) S h , ω ± is proven, then the index iteration theories of Bott and Long are also valid for the discrete case, and those can be also applied to the study of the symplectic algorithm.