SHARP DETERMINANTS

We introduce a sharp trace Tr#M and a sharp determinant Det#(1 - zM) f or an algebra of operators M acting on functions of bounded variation on the real line. We show that the zeroes of the sharp determinant des cribe the discrete spectrum of M. The relationship with weighted zeta functions of interval maps and Milnor-Thurston kneading determinants i s explained. This yields a result on convergence of the discrete spect rum of approximated operators.