Semi-classical trace formula and clustering of eigenvalues for Schrodinger operators

This paper is devoted to certain semi-classical asymptotics of a Schrodinger type operator A(h) in the vicinity of a regular value E of its principal symbol a(0)(x, xi). We investigate the semi-classical behaviour of the number N-E+rh,N-c(h) Of all eigenvalues lambda(j)(h) of A(h) situated in the interval [E+rh-ch, E+rh+ch], where the energy shift parameter r and the size constant c > 0 are both bounded. The behaviour of N-E+rh,N-c(h) for Small h depends on an oscillating term Q(h, r) which is related to the periodic trajectories of the Hamiltonian vector field H-a0 on the energy hypersurface Sigma = {(x, xi) : a(0)(x, xi) = E}. If Q(h, r) is uniformly continuous in r for any 0 < h less than or equal to h(0), we obtain asymptotics of the counting function N-E+rh,N-c(h) as h tends to zero. On the other hand, the points of discontinuity of Q(h, r) in r may give rise to a clustering of eigenvalues of A(h) near the energy level E. Such jumps of the function Q in r are described in terms of a suitable quantization condition. In particular, if a(0) is analytic in a neighborhood of Sigma and the energy surface is connected and of contact type we obtain a complete description of the asymptotics of N-E+rh,N-c(h) Moreover, we obtain a new semi-classical trace formula giving for any p(tau) is an element of S(R) with Fourier transform (p) over cap(t) is an element of C-0(infinity)(R) the asymptotics of Sigma(lambda j(h)less than or equal to lambda) p(E-lambda(j)(h)/h) in terms of certain dynamic and topological characteristics of the periodic trajectories of H-a0 on Sigma without any additional clean intersection assumptions. (C) Elsevier, Paris.