Chaotic wave functions and exponential convergence of low-lying energy eigenvalues

We suggest that low-lying eigenvalues of realistic quantum many-body hamiltonians, given, as in the nuclear shell model, by large matrices, can be calculated, instead of the full diagonalization, by the diagonalization of small truncated matrices with the exponential extrapolation of the results. We show numerical data confirming this conjecture. We argue that the exponential convergence in an appropriate basis may be a generic feature of complicated ("chaotic") systems where the wave functions are localized in this basis.