We introduce a sharp trace \(\text{Tr}^\# {\mathcal M}\) and a sharp determinant \(\text{Det}^\#(1- z{\mathcal M})\) for an algebra of operators \(\mathcal M\) acting on functions of bounded variation on the real line. We show that the zeroes of the sharp determinant describe the discrete spectrum of \(\mathcal M\). The relationship with weighted zeta functions of interval maps and Milnor-Thurston kneading determinants is explained. This yields a result on convergence of the discrete spectrum of approximated operators.