Geometric Optics and the Bottom of the Spectrum

The continuous spectrum, near its innmum, of the Lapla-cian for certain classes of`doubly-warped' complete Riemannian met-rics on the interiors of compact manifolds with boundary is examined using geometric optics. General conditions on the geometry of an algebra of vector elds leading to a calculus of pseudodiierential operators are described and, once veriied in this particular case, the associated notion of wavefront set can be used to show the existence of an absolute scattering matrix. This operator is conjectured to be a Fourier integral operator; the connrmation of this conjecture is discussed in the special case of a `double-zero' metric where the brations are trivial.