The diffractive wave trace on manifolds with conic singularities

Abstract

Abstract. Let (X, g) be a compact manifold with conic singularities. Taking ∆g to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group e −it √ ∆g arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points. IntroductionIn this paper, we consider the trace of the half-wave group U(t) def = e −it √ ∆g on a compact manifold with conic singularities (X, g). Our main result is a description of the singularities of this trace at the lengths of closed geodesics undergoing diffractive interaction with the cone points. Under the generic assumption that the cone points of X are pairwise nonconjugate along the geodesic flow, the resulting singularity at such a length t = L has the oscillatory integral representationwhere the amplitude a is to leading orderas |ξ| → ∞ and the index j is cyclic in {1, . . . , k}. Here, n is the dimension of X and k the number of diffractions along the geodesic, and χ is a smooth function supported in [1, ∞) and equal to 1 on [2, ∞). L 0 is the primitive length, in case the geodesic is an iterate of a shorter closed geodesic. The product is over the diffractions undergone by the geodesic, with D αj a quantity determined by the functional calculus of the Laplacian on the link of the j-th cone point Y αj , the factor Θ − 1 2 (Y αj → Y αj+1 ) is (at least on a formal level) the determinant of the differential of the flow between the j-th and (j + 1)-st cone points, and m γj is the Morse index of the geodesic segment γ j from the j-th to (j + 1)-st cone points. All of these factors are described in more detail below.To give this result some context, we recall the known results for the Laplace- is a well-defined distribution on R t . Moreover, it satisfies the "Poisson relation": it is singular only on the length spectrum of (X, g),L is the length of a closed geodesic in (X, g)} .(This was shown independently by Chazarain [Cha74]; see also [CdV73].) Subject to a nondegeneracy hypothesis, the singularity at the length t = ±L of a closed geodesic has a specific leading form encoding the geometry of that geodesic-the formula involves the linearized Poincaré map and the Morse index of the geodesic. The proofs of these statements center around the identificationwhere∆g is (half of) the propagator for solutions to the wave equation on X; in particular, U(t) is a Fourier integral operator.In this paper, we prove an analogue of the Duistermaat-Guillemin trace formula on compact manifolds with conic singularities (or "conic manifolds"), generalizing results of Hillairet [Hil05] from the case of flat surfaces with conic singularities. We again consider the trace Tr U(t), a spectral invariant equal to ∞ j=0 e −itλj . The Poiss...

A Quantitative Vainberg Method for Black Box Scattering

“…To prove Theorem 2, we use an argument similar to that used to prove a resonance expansion in [3,Theorem 2.7,4.42], [17].…”

“…For each component of Y , Y α , let −∆ α denote the Laplacian on Y α with respect to the metric h α := h| Yα . Then define the operators as in [4],…”

“…So that we may consider many different types of perturbation of the Euclidean Laplacian at once, we work in the black box setting originally developed by Sjöstrand-Zworski in [13]. In particular, the results apply to scattering in the presence of conic points [1,4], by delta potentials [5,6], by bounded obstacles etc.…”

“…The main application of our results is to scattering on manifolds with conic points. In [1] and [4] the Baskin-Wunsch and Ford-Wunsch respectively analyze the singularities of the wave group and the wave trace. We use these results together with Theorems 1 and 3 to give a generically optimal bound on the size of the resonance free region for scattering on a manifold with conic singularities.…”

The Neumann Isospectral Problem for Trapezoids

“…. There has also been a lot of research on the contribution of diffractive geodesics to the wave trace; see Hi05], Ford-Wunsch [FoWu14], and more recently Hassell-Ford-Hillairet [FoHaHi15]. In the physics literature, we also note the works of Bogomolny-Pavloff-Schmit [BoPaSc] and Pavloff-Schmit [PaSc95].…”