# Low frequency asymptotics and local energy decay for the Schr{\"o}dinger equation

@inproceedings{Royer2021LowFA, title={Low frequency asymptotics and local energy decay for the Schr\{\"o\}dinger equation}, author={Julien Royer}, year={2021} }

Abstract. We prove low frequency resolvent estimates and local energy decay for the Schrödinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schrödinger operator with the resolvent of the free Laplacian. This gives the leading term for the developpement of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for…

## References

SHOWING 1-10 OF 27 REFERENCES

Local energy decay for the damped wave equation

- Mathematics, Physics
- 2014

Abstract We prove local energy decay for the damped wave equation on R d . The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range…

Local energy decay for several evolution equations on asymptotically euclidean manifolds

- Mathematics, Physics
- 2010

Let P be a long range metric perturbation of the Euclidean Laplacian on R^d, d>1. We prove local energy decay for the solutions of the wave, Klein-Gordon and Schroedinger equations associated to P.…

Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups

- Mathematics
- 2020

Abstract In this article, we prove Gevrey estimates of the resolvent near threshold zero for a class of second order elliptic operators satisfying a weighted coercive condition. As application, we…

Sharp resolvent and time-decay estimates for dispersive equations on asymptotically Euclidean backgrounds

- Mathematics, PhysicsDuke Mathematical Journal
- 2021

The purpose of this article is twofold. First we give a very robust method for proving sharp time decay estimates for the most classical three models of dispersive Partial Differential Equations, the…

LOCAL DECAY FOR THE DAMPED WAVE EQUATION IN THE ENERGY SPACE

- Physics, MathematicsJournal of the Institute of Mathematics of Jussieu
- 2016

We improve a previous result about the local energy decay for the damped wave equation on $\mathbb{R}^{d}$ . The problem is governed by a Laplacian associated with a long-range perturbation of the…

Large time behavior of solutions to Schrödinger equation with complex-valued potential

- Mathematics, Physics
- 2019

We study the large-time behavior of the solutions to the Schrodinger equation associated with a non-selfadjoint operator having zero energy eigenvalue and real resonances. Our results extend those of…

Local energy decay and smoothing effect for the damped Schrödinger equation

- Mathematics, Physics
- 2017

We prove the local energy decay and the smoothing effect for the damped Schrodinger equation on R^d. The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric.…

Low Frequency Estimates and Local Energy Decay for Asymptotically Euclidean Laplacians

- Physics, Mathematics
- 2010

For Riemannian metrics G on ℝ d which are long range perturbations of the flat one, we prove estimates for (− Δ G − λ −iε)−n as λ → 0, which are uniform with respect to ε, for all n ≤ [d/2] +1 in odd…

Spectral asymptotics in the semi-classical limit

- Mathematics
- 1999

Introduction 1. Local symplectic geometry 2. The WKB-method 3. The WKB-method for a potential minimum 4. Self-adjoint operators 5. The method of stationary phase 6. Tunnel effect and interaction…

Local decay of scattering solutions to Schrödinger's equation

- Mathematics
- 1978

The main theorem asserts that ifH=Δ+gV is a Schrödinger Hamiltonian with short rangeV, φεLcompact2 (IR3), andR>0, then ‖exp(iHt)ΠSφ‖L2(|x|<R)=O(t−1/2), ast→∞ where ΠS is projection onto the…