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EDIT. I am trying to read the article “Supersymmetry and Morse theory” by E. Witten (JDG 17 (1982)). This well known article applies some tools developed by physicists (e.g. path integrals) to ...

Let $H$ be a self-adjoint Hamiltonian and $H$ admits a decomposition into closed operators $D,D^*$, such that we have $H = D^*D$. I will now consider the one-dimensional case on a compact set: So ...