This IMA Volume in Mathematics and its Applications QUASICLASSICAL METHODS is based on the proceedings of a very successful one-week workshop with the same title, which was an integral part of the 1994-1995 IMA program on "Waves and Scattering." We would like to thank Jeffrey Rauch and Barry Simon for their excellent work as organizers of the meeting. We also take this opportunity to thank the National Science Foun­ dation (NSF), the Army Research Office (ARO) and the Office of Naval Research (ONR), whose financial support made the workshop possible. A vner Friedman Robert Gulliver v PREFACE There are a large number of problems where qualitative features of a partial differential equation in an appropriate regime are determined by the behavior of an associated ordinary differential equation. The example which gives the area its name is the limit of quantum mechanical Hamil­ tonians (Schrodinger operators) as Planck's constant h goes to zero, which is determined by the corresponding classical mechanical system. A sec­ ond example is linear wave equations with highly oscillatory initial data. The solutions are described by geometric optics whose centerpiece are rays which are solutions of ordinary differential equations analogous to the clas­ sical mechanics equations in the example above. Much recent work has concerned with understanding terms beyond the leading term determined by the quasi classical limit. Two examples of this involve Weyl asymptotics and the large-Z limit of atomic Hamiltonians, both areas of current research.

Tutorial Overviews.- h-pseudodifferential operators and applications: An introduction.- Semi-classical methods with emphasis on coherent states.- Workshop Research Papers.- Approximative theories for large Coulomb systems.- Semiclassical analysis for the Schrödinger operator with magnetic wells (after R. Montgomery, B. Helffer-A. Mohamed).- On the asymptotic distribution of eigenvalues in gaps.- Asymptotics of the ground state energy of heavy molecules in the strong magnetic field.- Local trace formulæ.- A proof of the strong Scott conjecture.- Lieb-Thirring inequalities for the Pauli operator in three dimensions.- Exact anharmonic quantization condition (in one dimension).