Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$
How generally is it true that this ${\rm Witten{-}Laplacian}_{s,p}$ is positive semi-definite?
And when the above is true are there examples of compact manifolds and (preferably ``usual") functions on them $p$ s.t we know the exact spectrum of the corresponding ${\rm Witten{-}Laplacian}_{s,p}$? (or at least a lower bounds on its smallest non-zero eigenvalue)
By "usual" functions I mean we do not assume that $f$ is Morse or that $f$ satisfies the confining or the Villani condition or such. And I happy to know if the above is known even for just spheres or AdS.
For ``nice" real valued functions $f$ on $(M,g)$ we have for the square of the gradient of $f$,
$\Vert {\nabla_g f} \Vert ^2 = g(\nabla_g f,\nabla_g f) = \sum_{j=1}^n \sum_{i=1}^n g^{ij} \partial_i f \partial_j f$
and the Laplacian of $f$ being,
$\nabla_g^2 f := \frac{1}{\sqrt{\det(g)}} \sum_{i,j=1}^n \frac{\partial }{\partial x_i} \left ( \sqrt{\det(g)} g^{ij} \frac{f}{\partial x_j}\right )$
where we define the metric as $g = [g_{ij}] = g \left ( \partial_{x_i}, \partial_{x_j} \right )$ and $g^{-1} = [g^{ij}]$.
Then the ``${\rm Witten{-}Laplacian}_{s,p}$" will be the operator mapping,
$$ C^2(M,g) \ni h \mapsto \left ( -s^2 \nabla_g^2 + \Vert{\nabla_g p}\Vert ^2 -s \nabla_g^2 p \right ) h \in C^2(M,g)$$
