Resolvent bounds for repulsive potentials

Technical Summary: Resolvent Bounds for Repulsive Potentials

Problem Statement
This paper addresses the establishment of limiting absorption resolvent bounds for the semiclassical Schrödinger operator $P(h) = -h^2\Delta + V(x)$ in dimensions $n \ge 3$. The primary focus is on potentials $V$ that are repulsive but may exhibit singularities at the origin and possess specific decay or boundedness properties at infinity. The authors aim to extend existing results, which were largely confined to dimension one or bounded potentials, to higher dimensions with potentials that may be singular as $r \to 0$ (e.g., repulsive Coulomb-type behavior) and satisfy a one-sided repulsivity condition.

A secondary objective is to apply these resolvent bounds to derive time-decay estimates for the weighted energy of solutions to the associated wave equation with short-range repulsive potentials and compactly supported initial data.

Methodology
The core analytical tool employed is the spherical energy method, a technique utilizing the separation of variables in polar coordinates. The approach relies on the classical identity relating the Laplacian to radial and angular components:
$$r^{\frac{n-1}{2}}(-\Delta)r^{-\frac{n-1}{2}} = -\partial_r^2 + r^{-2}\Lambda,$$
where $\Lambda = -\Delta_{S^{n-1}} + \frac{(n-1)(n-3)}{4}$. A crucial observation is that $\Lambda \ge 0$ on $L^2(S^{n-1})$ for $n \ge 3$, which allows the use of a relatively simple weight function. This contrasts with more general potentials where conjugation by an exponential phase $e^{\phi/h}$ is typically required, leading to exponential losses as $h \to 0$.

The proof strategy involves:

1. Defining a Spherical Energy Functional: A functional $F[u](r)$ is constructed involving the radial derivative and the potential term.
2. Distributional Derivatives: The authors compute the distributional derivative of a weighted version of this energy, $wF$, utilizing the bounded variation properties of the potential $V$ along radial lines.
3. Weight Selection: A specific weight function $w(r)$ is chosen to exploit the repulsivity condition (inequality 1.9), ensuring that the resulting terms in the energy estimate have a suitable lower bound.
4. Density Arguments: Estimates are first derived for smooth, compactly supported test functions and then extended to the resolvent operator via density arguments in $H^2(\mathbb{R}^n)$.
5. Complex Analysis: The Phragmén-Lindelöf principle is used to extend bounds from specific sectors of the complex plane to the entire domain $\mathbb{C} \setminus [0, \infty)$.

For the wave equation application, the authors combine the resolvent bounds with Duhamel's formula, the Fourier transform, and Plancherel's theorem. They utilize the $\lambda$-derivative of the resolvent to control the time decay of the weighted energy, requiring stronger short-range conditions on the potential to handle the singularity at the origin.

Key Contributions and Results

1. Weighted Limiting Absorption Resolvent Bounds (Theorem 1.1 & 1.2):

   • General Case: For $n \ge 3$ and potentials satisfying mild singularity conditions at the origin (allowing $V \sim r^{-\rho(n)}$) and repulsivity, the paper proves:
     $$\|(r+1)^{-s}(P(h)-z)^{-1}(r+1)^{-s}\| \le \frac{C}{h}$$
     for $z = E \pm i\varepsilon$ with $E > 0$.
   • Non-negative Case ($V^- = 0$): Stronger estimates are obtained for non-negative potentials:
     $$\|(r+1)^{-s}(P(h)-z)^{-1}(r+1)^{-s}\| \le \frac{C}{h|z|^{1/2}}$$
     and
     $$\|(r+1)^{-s_1}(P(h)-z)^{-1}(r+1)^{-s_2}\| \le \frac{C}{h^2}$$
     provided $s_1 + s_2 > 2$. These bounds are nearly optimal in the sense of the $h$ and $z$ dependencies.

2. Time Decay for Wave Equations (Theorem 1.6):

   • Applying the resolvent bounds to the wave equation $(\partial_t^2 - \Delta + V)u = 0$, the authors establish weighted energy decay. For initial data with compact support and potentials satisfying specific short-range conditions (including a repulsivity condition and decay at infinity), the weighted energy $E_s(t)$ decays as:
     $$E_s(t) \le C_s \langle t \rangle^{-2} E(0)$$
     where $\langle t \rangle = (1+|t|^2)^{1/2}$. This holds for $s$ satisfying specific dimension-dependent thresholds (e.g., $s > (n+3)/4$ for $n \neq 8$).

3. Handling Singularities:
   The paper explicitly handles potentials with singularities at the origin (e.g., $V \sim r^{-\rho}$), a feature not covered by previous bounds for bounded potentials in higher dimensions. The authors note that the dimension $n=2$ remains open due to the strong negative singularity of the effective potential $-1/(4r^2)$ as $r \to 0$, which prevents the non-negativity of $\Lambda$.

Significance and Claims
The authors position this work as an extension of the results by Christiansen and Datchev [ChDa21], who established similar bounds for bounded, repulsive potentials in one dimension (and on the half-line). The novelty of this paper lies in:

• Extending these bounds to dimensions $n \ge 3$.
• Accommodating potentials that are singular at the origin (specifically allowing for repulsive Coulomb-like behavior).
• Demonstrating that the positivity of the potential (or the absence of a negative part $V^-$) is a sufficient condition to rule out threshold obstructions (resonances or eigenvalues at zero energy), thereby enabling the $O(h^{-2})$ resolvent bound and the resulting $O(t^{-2})$ energy decay.

The paper claims that the derived estimates are "nearly optimal" regarding the dependence on the semiclassical parameter $h$ and the spectral parameter $z$, as justified by counter-examples in the free case ($V=0$) presented in Appendix D. The authors also note that the restriction on the singularity strength $\rho(n)$ and the specific short-range decay conditions for the wave decay application are artifacts of their current approach (specifically the use of Hilbert-Schmidt norms for low-frequency analysis) and suggest that more singular potentials might be treatable with different methods.

The work does not propose new experimental setups or future applications beyond the mathematical derivation of these bounds and their immediate application to wave decay theory. It serves as a rigorous analytical foundation for understanding the spectral and scattering properties of repulsive Schrödinger operators in higher dimensions.