Let $M$ be an n-dimensional closed oriented Riemannian manifold.

For any $e\in\Gamma(TM)$, let $e^{*}\in\Gamma(T^{\ast}M)$ corresponds to e via $g^{TM}$.

Let $\hat{c}(e)$ be the Clifford operator acting on the exterior algebra bundle $\wedge^{*}(T^{\ast}M)\otimes\mathbb{C}$ defined by

$$\hat{c}(e)=e^{*}\wedge+i_{e}$$

where $e^{*}\wedge$ and $i_{e}$ are the standard notation for exterior and interior multiplications.

We known there is a de Rham-Hodge operator $$D=d+d^{*}.$$

The usually deformation of de Rham-Hodge operator by a vector field $V\in\Gamma(TM)$ is

$$D_{T}=d+d^{*}+T\hat{c}(V)$$

I want to discuss the deformation of de Rham-Hodge operator by two vector field

$X,Y\in \Gamma(TM)$, defined by

$$D_{T}=d+d^{*}+T(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))$$ for any $T\in\mathbb{R}$.

We known $$D_{T}^{2}=D^{2}+TD(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))+T^{2}(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))$$

I have two questionss:

1、Is the $D_{T}^{2}$ a $Schr\ddot{o}dinger$ type operator?

2、How to discuss the operator $D_{T}^{2}$ in "local"? I mean in small neighborhoods of zeros of $(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))$.

Remark:The notations come form the book 'Lectrues on Chern-Weil theory and Witten deformations' by Weiping Zhang.