[Paper Review] Dispersive estimates for Schroedinger operators: A survey

Wilhelm Schlag|ArXiv.org|Jan 3, 2005

Advanced Mathematical Physics Problems49 references5 citations

TL;DR

This survey presents recent advances in dispersive estimates for Schrödinger operators with time-independent and time-dependent potentials, focusing on $ L^1 \to L^\infty $ decay rates of the propagator $ e^{itH}P_c $. It establishes that under suitable decay and spectral conditions—such as absence of zero-energy resonances or eigenvalues—the decay rate $ |t|^{-d/2} $ holds, extending classical free-particle results and enabling Strichartz estimates via interpolation.

ABSTRACT

We present some old and new results on dispersive estimates for Schroedinger equations.

Motivation & Objective

• To summarize recent progress in dispersive estimates for Schrödinger operators with general potentials, particularly in the presence of bound states and zero-energy resonances.
• To clarify the role of spectral conditions—such as the absence of zero-energy eigenvalues or resonances—in determining the sharp $ L^1 \to L^\infty $ decay rate of the propagator.
• To unify and present techniques from perturbative and non-perturbative approaches, including resolvent expansions and weighted $ L^2 $ estimates.
• To extend dispersive estimates to time-dependent potentials, including periodic and charge-transfer models, and to connect them to scattering theory and Strichartz estimates.

Proposed method

• Use of asymptotic expansions of the resolvent $ (H - z^2)^{-1} $ near zero energy in odd dimensions, expressed as $ z^{-2}A_{-2} + z^{-1}A_{-1} + A_0 + zA_1 + O(z^2) $, to analyze long-time behavior of the propagator.
• Application of the Fredholm alternative and mapping properties of the free resolvent $ (-\triangle + i0)^{-1} $ on weighted $ L^2 $ spaces to control the behavior at zero energy.
• Interpolation between $ L^2 $-conservation and $ L^1 \to L^\infty $ dispersive estimates to derive Strichartz estimates of the form $ \|e^{itH}P_c f\|_{L^q_t L^p_x} \leq C\|f\|_2 $ under the scaling condition $ \frac{2}{q} + \frac{d}{p} = \frac{d}{2} $.
• Analysis of time-dependent potentials via the Floquet formalism and the use of the time-evolution operator $ U(t,s) $, reducing the problem to the study of the Floquet operator $ \mathcal{U} = U(T,0) $.
• Use of the $ T^*T $ argument to derive Strichartz estimates from dispersive bounds, with attention to the failure of the endpoint $ q=2 $ in this approach.
• Application of weighted $ L^2 $ estimates, such as $ \|w e^{itH}P_c w f\|_2 \leq C|t|^{-3/2}\|f\|_2 $, with $ w(x) = \langle x\rangle^{-\sigma} $ or $ e^{-\rho\langle x\rangle} $, to handle large potentials in $ d=3 $.

Experimental results

Research questions

• RQ1 Under what conditions on the potential $ V $ does the Schrödinger propagator $ e^{itH}P_c $ satisfy the dispersive estimate $ \|e^{itH}P_c f\|_\infty \leq C|t|^{-d/2}\|f\|_1 $?
• RQ2 How do zero-energy resonances or eigenvalues affect the dispersive decay rate, and what modifications are needed in the estimate?
• RQ3 Can dispersive estimates be extended to time-dependent potentials such as periodic or moving potentials, and what spectral conditions ensure decay?
• RQ4 What is the relationship between dispersive estimates, Strichartz estimates, and asymptotic completeness in the context of scattering theory?
• RQ5 How do the spectral properties of the Floquet operator influence the long-time behavior and decay of solutions in time-periodic systems?

Key findings

• For $ d \geq 3 $ and $ V $ decaying like $ \langle x\rangle^{-2-\varepsilon} $, the absence of zero-energy resonances ensures the dispersive estimate $ \|e^{itH}P_c f\|_\infty \leq C|t|^{-d/2}\|f\|_1 $ holds.
• If zero is a resonance but not an eigenvalue, the decay rate deteriorates to $ |t|^{-1/2} $ in $ d=3 $, indicating a loss of decay compared to the free case.
• In $ d=3 $, the dispersive estimate $ \|w e^{itH}P_c w f\|_2 \leq C|t|^{-3/2}\|f\|_2 $ holds for exponentially or polynomially decaying weights $ w $, provided zero is neither an eigenvalue nor a resonance.
• For small time-dependent potentials in $ \mathbb{R}^3 $, such as quasi-periodic or trigonometric polynomials, dispersive estimates with $ t^{-3/2} $ decay are established under $ \|V\|_{\mathcal{K}} < \infty $.
• In charge-transfer models with moving potentials $ V_j(\cdot - v_j t) $, dispersive estimates in $ L^1 \cap L^2 \to L^2 \cap L^\infty $ are proven, enabling asymptotic stability results for $ N $-soliton solutions.
• For time-periodic potentials, the absence of discrete spectrum in the Floquet operator implies scattering and local $ L^2 $ decay on the orthogonal complement of bound states, as shown by Galtbayar, Jensen, and Yajima.