  Motivated by the theory of superconductivity and more precisely by the
problem of the onset of superconductivity in dimension two, many papers devoted
to the analysis in a semi-classical regime of the lowest eigenvalue of the
Schr\"odinger operator with magnetic field have appeared recently. Here we
would like to mention the works by Bernoff-Sternberg, Lu-Pan, Del
Pino-Felmer-Sternberg and Helffer-Morame and also Bauman-Phillips-Tang for the
case of a disc. In the present paper we settle one important part of this
question completely by proving an asymptotic expansion to all orders for
low-lying eigenvalues for generic domains. The word `generic' means in this
context that the curvature of the boundary of the domain has a unique
non-degenerate maximum.
