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. Considering a known matrix H with eigen values of $\epsilon_i$ (which we don't want to calculate directly by solving eigen-value problem) How I can find an approximate for following according to matrix elements:

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

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

When C $\to \infty$ the answer is obvious

$\lim_{C \to \infty} \Sigma_{(\epsilon_i < C > )}(\epsilon_i) = trace(H)$