Bar codes of persistent chomology and arrhenius law for p-forms/ D. Le Peutrec, F. Nier & C. Viterbo

Includes bibliography

Introduction -- Boundary Witten Laplacians -- Local Problems -- Rough estimates for several "vritical Values" -- Singular values -- Accurate analysis with N "Critical values"-- Corollaries of theorem 6.3 -- Generalization -- Applications -- Broadening the scope

The present work shows that counting or computing the small eigenvalues of the Witten Laplacian in the Semi- classical limit can be done without assuming that the potential is a Morse function as the authors did in their previous article. In connection with persistent cohomology, we prove that the resealed logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the potential function. In particular, this proves that these quantities are stable in the uniform convergence topology of the space of continuous functions. Additionally, our analysis provides a general method for computing the sub exponential corrections in a large number of case.