Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery curvature conditions (i.e $\mathrm{Ricci_M} + \nabla^2 V \succeq cI_n$, for some $c > 0$). Consider the so-called Witten Laplacian $$ \Delta_\mu(f):=\Delta f - \langle \nabla V, \nabla f\rangle, $$ where $\nabla^2$ is the Hessian and $\Delta$ is the usual Laplacian and $\Delta$. It is known that $\Delta_\mu$ is a self-adjoint compact operator on $L^2(M,\mu)$. Let $0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \ldots \lambda_k \le \ldots$ be its eigenvalues.

Question. What is a good upper-bound for $\lambda_k$ in terms of $k$, $n$, $c$ (and perhaps $\lambda_1$, if that helps) ?

For example, the Gaussian case where $M=\mathbb R^n$ (euclidean) and $V(x) \equiv \|x\|^2/2+(n/2)\log(2\pi)$, would already be useful. Note that in this case, one has $c=1$.