Richard B. Melrose

The Atiyah-Patodi-Singer index theorem

Based on the lecture notes of a graduate course given at MIT, this sophisticated treatment leads to a variety of current research topics and will undoubtedly serve as a guide to further studies.

Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature

Abstract We give a detailed description of the Schwartz kernel of the resolvent of the Laplacian on a certain class of complete Riemannian manifolds with negative sectional curvature near infinity. These manifolds have compactifications as C∞ manifolds with boundary on which the metric is conformai to a C∞ metric (up to the boundary) with conformal factor ϱ−2, where ϱ is a defining function for the boundary. The modified resolvent R(ζ) = (Δ + s −2 ζ(ζ − n)) −1 defined and analytic in R (ζ > n is shown to extend to be meromorphic in the whole complex plane, when − s 2 is equal to the sectional curvature at infinity. For hyperbolic spaces obtained as quotients Γ⧹Hn + 1, where Γ is a discrete group of fractional linear transformations having a geometrically finite fundamental domain without cusps at infinity, s is constant. We then deduce from the existence of this meromorphic extension of the resolvent the existence of a similar extension for the Eisenstein series. The basic method used to produce a parametrix for the Laplacian is part of a more general construction, by G. Mendoza and the second author, but since this is not yet available the present paper has been made essentially self-contained.

TRANSFORMATION OF BOUNDARY PROBLEMS

II. PSEUDODIFFEI%ENTIAL OPERATORS i. Symbol spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 2. Operators on open sets . . . . . . . . . . . . . . . . . . . . . . . . . . 170 3. Definition on Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4. Kernels and adjoints . . . . . . . . . . . . . . ~ . . . . . . . . . . . . 176 5. Boundary values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6. Symbols and residual operators . . . . . . . . . . . . . . . . . . . . . . 187 7. Composition and el]iptieity . . . . . . . . . . . . . . . . . . . . . . . . 194 8. Wavefront set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9. Normal regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 i0. L ~ estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle

Abstract The uniform asympotic behavior of the scattering amplitude near the forward peak, in the case of classical scattering of waves by a convex obstacle, is derived. A microlocal model is obtained for the scattering operator. This is achieved by use of a parametrix for diffractive boundary problems and by a new study of a class of Fourier integral operators, those with folding canonical relations. A crucial ingredient consists of putting a Fourier integral operator with folding canonical relation into a normal form. The analysis also gives the asymptotic behavior of the normal derivative of the scattered wave on a neighbourhood of the shadow boundary, thus providing a corrected version of the Kirchhoff approximation.

The Poisson Summation Formula for Manifolds with Boundary

for E > 0. The sum on the right is over all closed geodesics, y; T, is the period of y, T,,* the primitive period of y, P, the Poincare map around y and R is in&, , The formula says that the distributional function of t defined by the sum on the left is equal to the sum on the right module locally I;,-summable functions, [3] and [4]. Suppose now that X is with boundary. Let A, , A, ,.,. be the eigenvalues for the boundary problem

The Propagation of Singularities along Gliding Rays.

SummaryLetP be a second-order differential operator with real principal symbol and fibre-simple characteristics on a manifold with boundary non-characteristic forP. LetB be a differential operator such that the boundary value problem (P, B) is normal and satisfies the Lopatinskii-Schapiro condition. The singularities of distributions,u, such thatP u is smooth on the boundary, near points at which the boundary is bicharacteristically convex are shown to propagate, in the boundary, only along the gliding rays, which are the leaves of the Hamilton foliation of the glancing surface. This analysis, combined with known results on diffraction, leads to a Poisson relation bounding the singular support of the Fourier transform of the Dirichlet spectral density for a compact Riemannian manifold with geodesically convex, or concave, boundary in terms of the geodesic length spectrum.

Analytic K-theory on manifolds with corners

Let Y be a compact manifold. In [ 1 ] Atiyah proposed an analytic delinition of the K-homology groups K,(Y), defined abstractly as the dual theory to the Atiyah-Hirzebruch K(cohomology)-theory. These ideas were developed by Kasparov [15] and Brown, Douglas, and Fillmore [lo]. Cycles for this analytic K-homology theory are “abstract elliptic operators,” i.e., bounded linear operators between Hilbert spaces satisfying additional conditions derived from the properties of elliptic (pseudo-) differential operators. In particular elliptic pseudodifferential operators between sections of vector bundles define such cycles. The homology class of the cycle depends only on the principal symbol and indeed only on the K-cohomology class it represents on the cotangent bundle. In fact it had already been shown, in essence, by Atiyah that the resulting map

Scattering theory and the trace of the wave group

In [9] Lax and Phillips discuss the analogue for classical scattering theories, in odd space dimensions, of the Poisson summation formula as generalized to compact manifolds by Chazarain [3], Colin de Verdi&e 141, and Duistermaat and Guillemin [5], and further generalized to manifolds with boundary in [ 1, 71. However, in [9], this trace formula is only proved for comparatively large values of the time variable (see also Ralston [ 14 1, Bardos el al. [2]). This note is devoted to the proof of the same formula for all positive time, for the case of scattering by a smooth compactly supported potential. If N(A) is the number of poles, ]A,] 3) then, for every E > 0 there exists C, > 0 such that

Asymptotics of Solutions of the Wave Equation on de Sitter-Schwarzschild Space

Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.

Weyl asymptotics for the phase in obstacle scattering

Soit O⊂R n , pour n≥3 impair, un obstacle lisse. On montre que la formule asymptotique de Weyl est vraie pour la phase de diffusion