Semiclassical diffraction by conormal potential singularities

Abstract

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 .

The radiation field on product cones

“…As in Melrose-Vasy-Wunsch [MVW08] and Gannot-Wunsch [GW18], we can arrange that a is well-localized: Lemma 6.9. Given any neighborhood U of q 0 ∈ Ḣ in b Ṡ * M and any β > 0, there are δ > 0 and c 1 > 0 so that a is supported in U for all 0 < δ < δ 0 .…”

A semiclassical approach to geometric X-ray transforms in the presence of convexity

“…More singular (with homogeneity 1) and thus delicate coisotropic algebras, corresponding to hypersurfaces, were introduced by Sjöstrand and Zworski [9]. (Though it was used for a different purpose and from a different perspective, the work [2] of Gannot and Wunsch introduced semiclassical paired Lagrangian distributions to extend the work of de Hoop, Uhlmann and Vasy [1] from the non-semiclassical setting, and this relates closely to 1-homogeneous 2-microlocalization at a coisotropic.) The 'blown-down' adjective refers to the fact that from this blow-up perspective, the standard semiclassical behavior (i.e.…”

“…We write, relative to our convex foliation and some coordinates, denoted by y, along the level sets, β = (x, y, λ, ω), so we write tangent vectors as λ ∂ x + ω ∂ y , and use γ (1) to denote the x component of γ, and similarly γ (2) to denote the y component of γ to avoid confusion. Then we have (3.2)…”

“…x,y,λ,ω (t)) = x + λt + α(x, y, λ, ω)t 2 + t 3 Γ (1) (x, y, λ, ω, t), y + ωt + t 2 Γ (2) (x, y, λ, ω, t) with Γ (1) , Γ (2) smooth functions of x, y, λ, ω, t, α a smooth function of x, y, λ, ω, and α(x, y, 0, ω) ≥ C > 0 by the concavity from the super-level sets hypothesis; see [14, Section 3] and [12, Proof of Proposition 4.2]. For us the relevant regime will be λ small; we shall restrict to an arbitrarily small neighborhood of λ = 0 via the semiclassical localization.…”

Semiclassical propagation through cone points

“…In the present paper we give an alternative proof which in particular avoids the use of a mixed calculus; see also Remark 1.5. We also mention that Gannot-Wunsch [GW18] analyzed the diffraction by conormal potentials in the semiclassical setting using direct commutator methods involving paired Lagrangian distributions, inspired by [dHUV15].…”