Is there any analytic expression for summation of eigen-values of a tri-diagonal matrix which are smaller than a constant value? Or even a rough approximation for it. How about case of a general matrix. i.e. if we have a matrix H then with eigen values of $\epsilon_i$ (which we don't want to calculate directly) What I need is following.

$\Sigma_{(\epsilon_i < C )}(\epsilon_i) = \Sigma_i(\epsilon_i * \Theta(C - \epsilon_i))$

Where $\Theta$ is step function and C is a constant.