Exponential local energy decay of solutions to the wave equation with $L^\infty$ electric and magnetic potentials

Technical Summary: Exponential Local Energy Decay of Solutions to the Wave Equation with $L^\infty$ Electric and Magnetic Potentials

Problem Statement
This paper investigates the long-time behavior of solutions to the wave equation associated with the magnetic Schrödinger operator $P = (i\nabla + b(x))^2 + V(x)$ in $\mathbb{R}^d$ ($d \ge 2$). The study focuses on two primary scenarios:

1. The whole space $\mathbb{R}^d$ ($d \ge 3$) with both electric ($V$) and magnetic ($b$) potentials.
2. The exterior of a non-trapping obstacle $O$ ($d \ge 2$) with an electric potential $V$, where the magnetic potential is identically zero ($b \equiv 0$).

The potentials are assumed to have $L^\infty$ regularity and satisfy an exponential decay condition at infinity: $|V(x)| + |b(x)| \le C e^{-c\langle x \rangle}$. The primary objective is to establish conditions under which the local energy of solutions to the wave equation decays exponentially in time. A critical challenge addressed is the behavior of low frequencies, which typically obstruct exponential decay unless specific spectral conditions (absence of eigenvalues or resonances at zero) are met or low frequencies are suitably cut off.

Methodology
The authors employ a strategy centered on deriving sharp resolvent estimates for the operator $P$ and utilizing these estimates to analyze the time evolution of the wave equation. The methodology proceeds through several key stages:

1. Carleman Estimates: The foundation of the analysis is the derivation of Carleman estimates for the free Laplacian on $\mathbb{R}^d$. These estimates are constructed using specific weight functions $\omega(r)$ and $\phi(r)$ designed to handle both medium and high frequencies. These estimates are crucial for obtaining limiting absorption bounds for the resolvent of the free operator.
2. Resolvent Bounds via Perturbation: Using the Carleman estimates and resolvent identities, the authors extend the bounds to the perturbed operator $P$.
   • For the whole space case ($d \ge 3$), they handle $L^\infty$ magnetic and electric potentials.
   • For the exterior domain case, they utilize a "black box" approach combined with a resolvent remainder argument. This allows them to transfer the known high-frequency non-trapping bound of the Dirichlet Laplacian to the perturbed operator, provided the magnetic potential vanishes.
3. Meromorphic Continuation: A pivotal step involves establishing the meromorphic continuation of the weighted resolvent $\mu(P - \lambda^2)^{-1}\mu$ (where $\mu(x) = e^{-c\langle x \rangle/2}$) from the lower half-plane to a region including the real axis. The exponential decay of the potentials is essential here, as it ensures the perturbation terms are compact operators, allowing the application of the Analytic Fredholm Theorem.
4. Low-Frequency Analysis: The paper derives specific bounds for the resolvent near zero frequency. In dimensions $d \ge 5$, under the assumption $V \ge 0$, the authors prove that the weighted resolvent remains bounded near zero without requiring a cutoff, provided zero is not an eigenvalue or resonance.
5. Time Decay via Resolvent Estimates: Finally, the authors apply the derived resolvent estimates (including bounds on their derivatives with respect to $\lambda$) to the solution formula of the wave equation. By utilizing a cutoff function $\psi_{\delta, t}$ that depends on time $t$ (or a Gevrey class function independent of $t$ with slightly weaker decay), they convert the resolvent bounds into exponential decay estimates for the local energy.

Key Contributions and Results

• Sharp Resolvent Estimates: The paper proves sharp resolvent estimates for the magnetic Schrödinger operator with $L^\infty$ potentials. Specifically, it establishes that the weighted resolvent $\mu(P - \lambda^2)^{-1}\mu$ and its derivatives satisfy bounds of the form $C_k k! (|\lambda|+1)^{\ell-1}$ for all $\lambda$ away from zero, and uniformly near zero under specific spectral conditions.
• Exponential Local Energy Decay:
  • General Case: For any dimension $d \ge 2$ (with $d \ge 3$ for the magnetic case), the authors prove that the local energy decays exponentially ($O(e^{-c_1 t})$) if the low frequencies are cut off by a function $\psi_{\delta, t}$. Notably, this cutoff function must depend on time $t$ to maintain the exponential rate, as no analytic function can satisfy the required cutoff properties.
  • Odd Dimensions without Cutoff: In odd dimensions, if zero is neither an eigenvalue nor a resonance of $P$, the low-frequency cutoff is unnecessary. The local energy decays exponentially without any modification to the initial data's spectral support.
  • Exterior Domains: For exterior domains with non-trapping obstacles, exponential decay is proven for the Dirichlet realization of $-\Delta + V$ (with $b \equiv 0$) under the same spectral conditions.
• Handling $L^\infty$ Potentials: The method is specifically designed to handle $L^\infty$ coefficients, a significant generalization over previous works that often required smooth potentials. The flexibility of the method across different frequency ranges is highlighted as a novel feature.

Significance and Claims
The authors claim that this work provides the first result of its type demonstrating exponential local energy decay for wave equations with $L^\infty$ electric and magnetic potentials that decay exponentially at infinity, applicable to both odd and even dimensions.

• Novelty in Low Frequencies: The paper emphasizes that the ability to achieve exponential decay in odd dimensions without cutting off low frequencies (provided the spectral condition at zero is met) is a direct consequence of the detailed control over the resolvent derivatives down to zero frequency, a control established via the meromorphic continuation.
• Methodological Flexibility: The approach is noted for its suitability in handling non-smooth ($L^\infty$) coefficients and its adaptability to both whole space and exterior domain problems.
• Limitations: The authors explicitly note that the case of $d=2$ with a magnetic potential is not treated due to obstructions in the Carleman estimate near the origin (related to the effective potential in polar coordinates). Additionally, while they conjecture that similar results might hold in even dimensions with polynomial decay rates ($t^{-2d}$) without low-frequency cutoffs, this requires further control on resolvent derivatives not fully established in the current work.

In summary, the paper bridges the gap between resolvent theory for non-smooth magnetic Schrödinger operators and the time-decay properties of the associated wave equation, providing rigorous conditions for exponential energy decay in a broad setting.