In the 80s, E. Witten introduced a version of the Laplacian on p-forms, distorted by a function f. If a Morse function f is given on M = Rn (or on a compact Riemannian manifold M without a boundary) and if h is a positive constant, an analysis of the properties of small eigenvalues of the semiclassical Witten Laplacian ∆(f,h) was given by A. Bovier, M. Eckhoff,V. Gayrard, B. Helffer, M. Klein and F. Nier. The main theme of the book presented here is to extend the mentioned results to the case with boundary (a domain in Rn and its smooth boundary, or a compact connected oriented Riemannian manifold with a boundary) for 0-forms. An important part of the work consists of a construction of a Witten cohomology complex adapted to the case with boundary.

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