Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?

Question

I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula?

For example, let $f_i $ be the eigenfunctions of an operator $D$, i.e., $$Df_i=\lambda_if_i$$ with $\lambda_i$ increasing.

Is there any formula taking care of $$\sum_i \frac{L(f_i)\overline{ L'(f_i)}}{||f_i||^2_2 } g(\lambda_i)$$ where $L$ is a linear functional? ($g$ is a test function.)