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This paper concentrates on analyzing Witten deformation for a family ofnon-Morse functions parametrized by $T\in \mathbb{R}_+$, resulting in a novel,purely analytic proof of the gluing formula for analytic torsions in completegenerality. Intriguingly, the gluing formula in this article could bereformulated as the Bismut-Zhang theorem for non-Morse functions, and from theperspective of Vishik's theory of moving boundary problems, the deformationparameter $T$ parametrizes a family of boundary conditions. Our proof alsomakes use of an interesting connection between small eigenvalues of WittenLaplacians and Mayer-Vietoris sequences. Finally, these new techniques could beextended to analytic torsion forms and will play key roles in our future studyof the higher Cheeger-M\"uller theorem for general flat bundles.

Witten deformation for non-Morse functions and gluing formula for analytic torsions