We consider the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions, and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularising terms, of parametrices with prescribed b-wavefront set. Furthermore, in the context of quantum fields, we show a similar result for two-point functions satisfying a holographic Hadamard condition on the b-wavefront set.
We study quasinormal modes for massive scalar fields in Schwarzschildanti-de Sitter black holes. When the mass-squared is above the Breitenlohner-Freedman bound we show that for large angular momenta, , there exist quasinormal modes with imaginary parts of size exp(− /C). We provide an asymptotic expansion for the real parts of the modes closest to the real axis and identify the vanishing of certain coefficients depending on the dimension.
This paper considers boundary value problems for a class of singular elliptic operators which appear naturally in the study of asymptotically anti-de Sitter (aAdS) spacetimes. These problems involve a singular Bessel operator acting in the normal direction. After formulating a Lopatinskiǐ condition, elliptic estimates are established for functions supported near the boundary. The Fredholm property follows from additional hypotheses in the interior. This paper provides a rigorous framework for mode analysis on aAdS spacetimes for a wide range of boundary conditions considered in the physics literature. Completeness of eigenfunctions for some Bessel operator pencils is shown.
We establish propagation of singularities for the semiclassical Schrödinger equation, where the potential is conormal to a hypersurface. We show that semiclassical wavefront set propagates along generalized broken bicharacteristics, hence reflection of singularities may occur along trajectories reaching the hypersurface transversely. The reflected wavefront set is weaker, however, by a power of h that depends on the regularity of the potential. We also show that for sufficiently regular potentials, wavefront set may not stick to the hypersurface, but rather detaches from it at points of tangency to travel along ordinary bicharacteristics. 10 ORAN GANNOT AND JARED WUNSCH 3.3. Interaction with differential operators 3.4. Wavefront set and ellipticity 3.5. b-Calculus relative to an interior hypersurface 4. Bicharacteristics 4.1. The characteristic set 4.2. Hamilton flow 4.3. Generalized broken bicharacteristics 5. Propagation of singularities along GBBs 5.1. The elliptic region 5.2. The hyperbolic region 5.3. The glancing region 5.4. Proof of Theorem 1 6. Semiclassical paired Lagrangian distributions 6.1. Nested conormal distributions 6.2. Change of variables 6.3. Pseudodifferential operators with singular symbols 6.4. Homogeneous paired Lagrangian distributions 7. Diffractive improvements 7.1. Decomposing the potential 7.2. Elliptic estimates 7.3. Improvement at hyperbolic points 7.4. Improvement at glancing points Appendix A. Proof of Proposition 1.1 A.1. Plane wave solutions A.2. WKB solutions A.3. Wronskians References * (R kx ′ )). We now return to the assumption that µ < −k. It is well known (see e.g. [Tay1, Section 13.8]) that if s = r + α for some r ∈ N and α ∈ (0, 1), then C s * (R k ) agrees with the Hölder space C r,α (R k ). From this, we immediately obtain the following lemma. Lemma 2.1. If µ < −k, then there exists θ ∈ (0, 1] depending only on µ + k such that any u ∈ C −∞ c (R m ) of the form (2.1) satisfies |u(x) − u(y)| ≤ C(|x ′ − y ′ | θ + |x ′′ − y ′′ |) (2.2) for each x, y ∈ R m .
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