  An expression for the Green function G(E;x_1,x_2) of the Schroedinger
equation is obtained through the approximations of the path integral by n-fold
multiple integrals. The approximations to Re{G(E;x,x)} on the real E-axis have
peaks near the values of the energy levels E_{j}. The analytic and numerical
examples for one-dimensional and multi-dimensional harmonic and anharmonic
oscillators, and Poeschl-Teller potential wells, show that median values of
these peaks for approximate G(E;0,0) corresponds with accuracy of order 10% to
the exact values of even levels already in the lowest orders of approximation
n=1 and n=2, i.e. when the path integral is replaced by a line or double
integral. The weights of the peaks approximate the values of the squared
modulus of the wave functions at x=0 with the same accuracy.
