Witten-Helffer-Sjostrand Theory for a Generalized Morse Functions

In this paper, we extend the Witten-Helffer-Sj"{o}strand theory from Morse functions to generalized Morse functions. In this case, the spectrum of the Witten deformed Laplacian $\mathrm{\Delta }(t)$, for large t, can be seperated into the small eigenvalues (which tend to 0 as $t\to \mathrm{\infty }$), large and very large eigenvalues (both of which tend to $\mathrm{\infty }$ as $t\to \mathrm{\infty }$). The subcomplex ${\mathrm{\Omega }}_0^{\ast }(M,t)$ spanned by eigenforms corresponding to the small and large eigenvalues of $\mathrm{\Delta }(t)$ is finite dimensional. Under some mild conditions, it is shown that $({\mathrm{\Omega }}_0^{\ast }(M,t),d(t))$ converges to a geometric complex associated to the generalized Morse function as $t\to \mathrm{\infty }$.