I am trying to prove the corollary $5.15$ of this paper by Colding and Minicozzi on page $28$ (it is the page $782$ with respect to the numeration of the journal), which is The Eigenvalue Problem for the operator defined by Colding and Minicozzi on page $26$ (it is the page $780$ with respect to the numeration of the journal). The specific inner product which turns out the operator self-adjoint is also defined in the same page in the theorem $5.2$.
Gilbarg and Trudinger in their famous book about elliptic PDEs prove the Eigenvalue Problem for a self-adjoint operator in the form
$$Lu = D_i(a^{ij}D_j u + b^i u) - b^i D_i u + c u,$$
where $[a^{ij}]$ is symmetric.
This is on page $212$.
An important identity is
$$\mathcal{L}v = \Delta v - \frac{1}{2} \left\langle x,\nabla v \right\rangle = e^{\frac{|x|^2}{4}} \text{div} \left( e^{-\frac{|x|^2}{4}} \nabla v \right)$$
on page $18$ (it is the page $772$ with respect to the numeration of the journal) of the paper. This combined with
$$\text{grad} f = \sum_{i,j} g^{ij} \frac{\partial f}{\partial x_i} \partial_j,$$
$$\text{div} V = \sum_i V_{; i}^i = \sum_i \left\{ \frac{\partial V^i}{\partial x^i} + \sum_j \Gamma_{ij}^i V^j \right\}$$
and
the operator defined by Colding and Minicozzi, which I put here by convenience,
$$Lu = \Delta u + |A|^2 u + \frac{1}{2} u - \frac{1}{2} \left\langle x, \nabla u \right\rangle$$
give
$\begin{align} L H &= e^{\frac{|x|^2}{4}} \sum_i \left\{ \frac{\partial}{\partial x_i} \left( e^{-\frac{|x|^2}{4}} g^{ij} \frac{\partial H}{\partial x_j} \right) + \sum_j \Gamma_{ij}^i \left( e^{-\frac{|x|^2}{4}} \sum_k g^{jk} \frac{\partial H}{\partial x_k} \right) \right\} + |A|^2 H - \frac{1}{2} H \end{align}$
I thought that I need to solve the following equation
$$ \frac{\partial b^k}{\partial x_k} H = \sum_i \left\{ \sum_j \Gamma_{ij}^i \left( \sum_k g^{jk} \frac{\partial H}{\partial x_k} \right) \right\}$$
to find $b^k$ and then the Colding and Minicozzi's operator be written in the form given by Gilbarg and Trudinger, but I could not develop from here. I would like some help to rewrite the operator above as in the form given by Gilbarg and Trudinger.
Thanks in advance!