Morse index and Maslov-type index of the discrete Hamiltonian system

The uploaded paper proves, for the discrete Hamiltonian system with ω-boundary and h=1/N small, the equalities m^-_{h,ω}=mN+i_ω(γ_{d,h,N}), m^0_{h,ω}=ν_ω(γ_{d,h,N}), and m^+_{h,ω}=mN−i_ω(γ_{d,h,N})−ν_ω(γ_{d,h,N}) (Theorem 1.1), using a direct homotopy reduction to standard symplectic paths and explicit index computations for those models, plus a perturbation argument for the degenerate case. The functional A_{h,ω} and the discrete path γ_{d,h,N} are set up exactly as in the solver’s write-up, and the nullity identity m^0_{h,ω}=ν_ω(γ_{d,h,N}(1)) is established via a Floquet-type argument. Small-step conditions ensure symplecticity of the discrete step maps. All of this is stated explicitly in the paper’s definitions, lemmas, and the proof of Theorem 1.1 . The model’s solution reaches the same result but via a different route: it frames A_{h,ω} as a selfadjoint path L_h(s), identifies spectral flow with the Maslov-type index (à la CLM/Long), computes the baseline m^- at s=0, and concludes the formulas. This argument is standard and consistent with the paper’s objects and conventions; the only missing piece is a brief justification that the CLM spectral flow/Maslov index equality adapts to the discrete ω-boundary setup (reasonable in finite dimension). Hence both are correct, with substantially different methods.