[Paper Review] On the Schrödinger equation with potential in modulation spaces

Elena Cordero, Fabio Nicola|arXiv (Cornell University)|May 13, 2014

Mathematical Analysis and Transform Methods43 references4 citations

TL;DR

This paper establishes the well-posedness and time-frequency propagation of singularities for the Schrödinger equation with a Hamiltonian composed of a quadratic Weyl operator and a rough potential in modulation spaces. It introduces a new Gabor wave front set to characterize singularities and proves that the evolution operator is the composition of a metaplectic operator and a pseudodifferential operator with symbol in modulation spaces, ensuring optimal time-frequency localization of solutions.

ABSTRACT

This work deals with Schrödinger equations with quadratic and sub-quadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations. We shall give a representation of such evolution as the composition of a metaplectic operator and a pseudodifferential operator having symbol in certain classes of modulation spaces. About propagation of singularities, we use a new notion of wave front set, which allows the expression of optimal results of propagation in our context. To support this claim, many comparisons with the existing literature are performed in this work.

Motivation & Objective

• To analyze the Schrödinger equation with a quadratic Hamiltonian perturbed by a rough potential in modulation spaces.
• To establish well-posedness of the Cauchy problem for initial data in modulation spaces with polynomial weights.
• To introduce a new Gabor wave front set that captures optimal propagation of singularities in the presence of rough potentials.
• To characterize the evolution operator as a composition of a metaplectic operator and a pseudodifferential operator with symbol in modulation spaces.
• To compare the new wave front set with classical notions and validate its optimality through comparisons with existing literature.

Proposed method

• Represent the Schrödinger evolution operator as the composition of a metaplectic operator and a pseudodifferential operator with symbol in modulation spaces.
• Use the short-time Fourier transform (STFT) with a Gaussian window to define time-frequency localization and modulation spaces.
• Introduce a new Gabor wave front set $ WF^{p,r}_G $ to describe the propagation of singularities for rough symbols in $ M^{ ilde{ ho}}_{1 ensor v_s} $ with $ s > 2d $.
• Apply the theory of modulation spaces and Wiener algebras to control the regularity and decay of the STFT of solutions.
• Use the phase-space evolution $ ilde{\mathcal{A}}_t $ to describe the trajectory of wave packets and relate it to the metaplectic action.
• Establish boundedness and continuity of the evolution operator on weighted modulation spaces via estimates on the STFT and symbol classes.

Experimental results

Research questions

• RQ1 Under what conditions on the potential $ \sigma $ is the Schrödinger equation with quadratic Hamiltonian well-posed in modulation spaces?
• RQ2 How does the time-frequency localization of the solution evolve when the potential is rough and in a modulation space $ M^{ ilfty}_{1 ensor v_s} $?
• RQ3 Can a new wave front set concept capture the optimal propagation of singularities in the presence of non-smooth potentials?
• RQ4 What is the precise structure of the evolution operator in terms of metaplectic and pseudodifferential operators when the symbol lies in modulation spaces?
• RQ5 How does the Gabor wave front set relate to classical wave front sets and what advantages does it offer in the context of rough symbols?

Key findings

• The evolution operator $ e^{itH} $ is represented as the composition of a metaplectic operator and a pseudodifferential operator with symbol in modulation spaces, ensuring precise time-frequency localization.
• For $ u_0 \in M^p_{v_r} $ with $ |r| < \mu - 2 $, the Cauchy problem is well-posed when the potential $ \sigma \in M^{ ilde{\rho}}_{1 \tensor v_{\mu+1}} $, with $ \mu > 1 $.
• The Gabor wave front set satisfies $ WF^{p,r}_G(e^{itH}u_0) = \mathcal{A}_t(WF^{p,r}_G(u_0)) $, showing that singularities propagate along phase-space trajectories.
• The wave front set $ WF_G(e^{2\pi i x \xi_0}) $ is $ \{(x,\xi) : x \neq 0, \xi = 0\} $, and $ WF_G(e^{\pi i c |x|^2}) = \{(x,\xi) : x \neq 0, \xi = c x\} $, confirming consistency with phase-space dynamics.
• For $ \sigma \in M^\infty_{1 \tensor v_s} $ with $ s > 2d $, the propagation of singularities is controlled via $ WF^{p,r}_G(\sigma(x,D)u) \subset WF^{p,r}_G(u) $ for $ 0 < 2r < s - 2d $.
• The results generalize and improve upon earlier works by Hörmander and others, providing optimal propagation results in the context of rough potentials through the new Gabor wave front set.