[Paper Review] Decay for the wave and Schroedinger evolutions on manifolds with conical ends, Part II

Wilhelm Schlag, Avy Soffer+1 more|ArXiv.org|Jan 14, 2008

Advanced Mathematical Physics Problems9 references4 citations

TL;DR

This paper establishes sharp global-in-time dispersive estimates for the Schrödinger and wave equations on manifolds with conical ends, using a novel scattering analysis of one-dimensional Schrödinger operators with long-range potentials and non-stationary phase methods. The key result is accelerated local decay of the form $ \|w_\sigma e^{it\Delta_{\mathcal{M}}} Y_n f\|_{L^\infty} \lesssim t^{-(d+1)/2 - \sigma} \|w_\sigma^{-1} f\|_{L^1} $, where $ \sigma = \sqrt{2\mu_n^2 + (d-1)^2/4} - (d-1)/2 $, valid for $ n \neq 0 $ and $ d \geq 1 $, under non-resonant zero-energy conditions.

ABSTRACT

Global in time dispersive estimates for the Schroedinger and wave evolutions are obtained on manifolds with conical ends whose Hamiltonian flow exhibits trapping. This paper deals with the case of initial data with fixed "nonzero angular momentum".

Motivation & Objective

• To establish global-in-time dispersive estimates for the Schrödinger and wave evolutions on manifolds with conical ends, which exhibit trapping in the Hamiltonian flow.
• To extend the results of Part I by treating the non-resonant case at zero energy, excluding the special $ d=1, n=0 $ case already covered.
• To derive accelerated local decay rates for solutions with angular momentum $ n \neq 0 $, quantified by a weight $ w_\sigma(x) = \langle x\rangle^{-\sigma} $ with $ \sigma > 0 $.
• To develop a detailed scattering theory for one-dimensional Schrödinger operators with long-range potentials decaying as $ \xi^{-3} $, including asymptotic expansion of the Wronskian at zero energy.
• To apply non-stationary phase methods to oscillatory integrals arising from spectral projections, ensuring decay rates consistent with the geometric structure of the manifold.

Proposed method

• A scattering analysis of one-dimensional Schrödinger operators $ -\partial_\xi^2 + (\nu^2 - 1/4)\langle\xi\rangle^{-2} + U(\xi) $ with $ U^{(\ell)}(\xi) = O(\xi^{-3-\ell}) $ as $ \xi \to \pm\infty $, focusing on zero-energy resonance structure.
• Introduction of a zero-energy resonance notion for this class of operators, enabling classification of the non-resonant case treated in this paper.
• Asymptotic expansion of the Wronskian between outgoing Jost solutions as energy $ \to 0 $, crucial for controlling spectral projections.
• Estimation of oscillatory integrals via non-stationary phase method, exploiting cancellation in phase and amplitude to derive decay rates.
• Use of spectral projection formulas involving Jost solutions and Wronskians to express the evolution operators in terms of integrals over the continuous spectrum.
• Weighted $ L^1 \to L^\infty $ estimates using $ w_\sigma(x) = \langle x\rangle^{-\sigma} $, with $ \sigma = \sqrt{2\mu_n^2 + (d-1)^2/4} - (d-1)/2 $, to capture accelerated decay.

Experimental results

Research questions

• RQ1 What is the precise decay rate of the Schrödinger evolution $ e^{it\Delta_{\mathcal{M}}} $ on manifolds with conical ends when the initial data has non-zero angular momentum $ n \neq 0 $?
• RQ2 How does the presence of trapping in the Hamiltonian flow affect dispersive estimates, and can accelerated local decay still be achieved despite trapping?
• RQ3 What is the role of zero-energy resonance in the spectral theory of Schrödinger operators on conical manifolds, and how does its absence simplify the analysis?
• RQ4 Can non-stationary phase methods be effectively applied to oscillatory integrals arising from spectral projections on non-Euclidean geometries with long-range potentials?
• RQ5 What is the optimal weight $ w_\sigma $ that captures the accelerated decay of solutions, and how is $ \sigma $ related to the geometry and spectral data of the base manifold $ \Omega $?

Key findings

• For $ n \neq 0 $, the Schrödinger evolution satisfies the dispersive estimate $ \|w_\sigma e^{it\Delta_{\mathcal{M}}} Y_n f\|_{L^\infty} \lesssim t^{-(d+1)/2 - \sigma} \|w_\sigma^{-1} f\|_{L^1} $ for all $ t \geq 1 $, with $ \sigma = \sqrt{2\mu_n^2 + (d-1)^2/4} - (d-1)/2 $.
• The wave evolution satisfies $ \|w_\sigma e^{\pm it\sqrt{-\Delta_{\mathcal{M}}}} Y_n f\|_{L^\infty} \lesssim t^{-d/2 - \sigma} \left( \|f' / w_\sigma\|_{L^1} + \|f / w_\sigma\|_{L^1} \right) $, with the same $ \sigma $, for $ t \geq 1 $.
• The decay rate is accelerated compared to the Euclidean case due to the $ \sigma > 0 $ weight, which has no analogue in flat space and vanishes for $ n = 0 $.
• The method relies on a non-resonant zero-energy scattering theory for one-dimensional Schrödinger operators with $ U(\xi) = O(\xi^{-3}) $, where the Wronskian has a controlled asymptotic expansion at zero energy.
• Non-stationary phase estimates are used to control oscillatory integrals arising from spectral projections, with cancellation exploited via $ W_{\nu}(-\lambda) = -W_{\nu}(\lambda) + O(1) $ as $ \lambda \to \infty $.
• The results are uniform in $ \sigma \in \left[0, \nu(d,n) - \frac{d-1}{2}\right] $, and the constants depend on $ n $, the manifold $ \mathcal{M} $, and $ \sigma $.