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