[Paper Review] Decay estimates for the Schroedinger evolution on asymptotically conic surfaces of revolution I

Wilhelm Schlag, Avy Soffer+1 more|ArXiv.org|Aug 3, 2006

Advanced Mathematical Physics Problems10 references3 citations

TL;DR

This paper establishes a pointwise $ L^∞ $ decay estimate for the Schrödinger evolution on asymptotically conic, symmetric surfaces of revolution, proving $ \|e^{it\Delta_{\mathcal{S}}}f\|_{L^\infty(\mathcal{S})} \lesssim |t|^{-1}\|f\|_{L^1(\mathcal{S})} $ for radial initial data. The analysis combines spectral theory, WKB asymptotics, and oscillatory integral estimates to handle the trapped geodesic at the cone point, extending dispersive bounds to non-trapping manifolds with geometric trapping.

ABSTRACT

We establish a dispersive estimate (with a decay of 1/t), valid for all times, for the Schroedinger evolution on a non-compact 2-dimensional manifold with a trapped geodesic.

Motivation & Objective

• To establish dispersive $ L^\infty $ decay estimates for the Schrödinger evolution on non-compact 2D manifolds with trapped geodesics, where standard dispersion fails.
• To extend the free-space dispersive bound $ \|e^{it\Delta}f\|_{L^\infty} \lesssim |t|^{-1} \|f\|_{L^1} $ to asymptotically conic surfaces of revolution with radial initial data.
• To analyze the impact of geometric trapping—specifically, a trapped geodesic at the cone point—on the dispersive behavior of the Schrödinger flow.
• To develop a spectral approach using Jost solutions and Wronskian asymptotics to control the Schrödinger propagator in the presence of trapping.

Proposed method

• The analysis focuses on surfaces of revolution defined by $ r(x) = |x|h(x) $ with $ h(x) = 1 + O(x^{-2}) $, asymptotically conic and symmetric.
• The Laplace-Beltrami operator $ \Delta_{\mathcal{S}} $ is analyzed via separation of variables, reducing the problem to radial Schrödinger operators on the real line.
• Jost solutions $ f_{\pm}(\xi, \lambda) $ are constructed for the radial Schrödinger equation with potential $ V(\xi) $, and their asymptotic behavior is analyzed for large $ \lambda $.
• The Wronskian $ W(\lambda) $ of the Jost solutions is computed and shown to satisfy $ W(\lambda) = -2i\lambda + O(1) $ as $ |\lambda| \to \infty $, enabling control of the spectral measure.
• Oscillatory integrals involving the spectral measure are estimated using stationary phase and integration by parts, with critical points at $ \lambda_0 = -\frac{\xi \pm \xi'}{2t} $.
• The key estimate relies on bounding the $ L^\infty $ norm of the evolution kernel via $ (\langle\xi\rangle\langle\xi'\rangle)^{-1/2} $ and $ t^{-1} $ decay from phase stationarity and spectral bounds.

Experimental results

Research questions

• RQ1 Can dispersive $ L^\infty $ decay estimates be established for the Schrödinger evolution on asymptotically conic surfaces with trapped geodesics?
• RQ2 How does geometric trapping at the cone point affect the decay rate of the Schrödinger propagator compared to the free case?
• RQ3 What spectral and oscillatory integral techniques are effective in controlling the Schrödinger evolution on non-trapping manifolds with asymptotically conic geometry?
• RQ4 To what extent does radial symmetry simplify the analysis of dispersive estimates on such manifolds?

Key findings

• The paper establishes the sharp $ |t|^{-1} $ decay estimate $ \|e^{it\Delta_{\mathcal{S}}}f\|_{L^\infty(\mathcal{S})} \lesssim |t|^{-1}\|f\|_{L^1(\mathcal{S})} $ for radial initial data on asymptotically conic, symmetric surfaces of revolution.
• The decay rate $ t^{-1} $ is consistent with the free 2D Schrödinger operator, despite the presence of a trapped geodesic, due to the specific asymptotic geometry of the surface.
• For $ r(x) = \langle x\rangle^\alpha $, the decay rate is $ t^{-\frac{1}{2}(1+\alpha)} $ when $ 0 < \alpha < 1 $, and $ t^{-1} $ when $ \alpha \geq 1 $, reflecting the influence of the asymptotic cone angle.
• The Wronskian of the Jost solutions satisfies $ W(\lambda) = -2i\lambda + O(1) $, which is essential for controlling the spectral measure and proving the decay estimate.
• The method relies on precise asymptotics of Jost solutions and careful estimation of oscillatory integrals using stationary phase, even when the critical point $ \lambda_0 $ is large.