Random Schrödinger operators on manifolds and abstract bounds for multiplier-type operators

Technical Summary: Random Schrödinger Operators on Manifolds and Abstract Bounds for Multiplier-Type Operators

Problem Statement
The paper investigates the spectral properties of random Schrödinger operators $H_{\omega, \lambda} = -\Delta_g + V_{\omega, \lambda}$ defined on a closed (compact, boundaryless) Riemannian manifold $(M, g)$ of dimension $d \geq 2$. The potential $V_{\omega, \lambda}$ is of Anderson type, constructed from a sum of localized "bumps" at scale $\lambda^{-1}$ with random coefficients $\omega_j$ (i.i.d. symmetric Bernoulli or centred normalized Gaussians).

The primary objective is to establish high-probability bounds on the location of the eigenvalues of $H_{\omega, \lambda}$ up to energy $\lambda^2$. Specifically, the authors aim to quantify how close the spectrum of the perturbed operator remains to the spectrum of the unperturbed Laplacian $-\Delta_g$. The study seeks to demonstrate that randomization yields a "square-root cancellation" gain compared to deterministic spectral inclusion bounds, which typically depend on the $L^\infty$ or $L^q$ norms of the potential without probabilistic improvement.

Methodology
The proof strategy relies on a two-step approach: first, establishing a general abstract principle regarding the operator norm of randomized multipliers, and second, applying this principle to specific spectral and resolvent operators on manifolds.

1. Abstract Operator-Theoretic Principle:
   The core innovation is a general framework showing that randomization improves norm bounds for operators of the form $T_1^* V_\omega T_2$, where $V_\omega$ is a randomized multiplication operator and $T_1, T_2$ are structured linear operators (e.g., spectral projectors, resolvents).

   • Discrete Setting: The authors formulate a discrete estimate (Theorem 1.2) where a function on a finite set is randomized over a partition. The proof utilizes a chaining argument inspired by Bourgain's entropy bounds, leveraging sub-Gaussian tail estimates (Dudley's inequality) and the geometry of the partition.
   • Continuous Setting: This is extended to Ahlfors $d$-regular metric measure spaces (Theorem 1.4). The randomization is performed on Voronoi cells associated with a maximal $r$-separated set. The authors introduce technical conditions such as "local constancy" and "uniform reparametrisation" to handle the continuous geometry, allowing the transfer of discrete probabilistic estimates to the manifold setting.

2. Application to Spectral Theory:
   The abstract bounds are applied to two specific cases relevant to the Schrödinger operator:

   • Spectral Projectors: Using Sogge's spectral cluster bounds for the operator $\Pi_\lambda = \mathbb{1}_{[\lambda, \lambda+1]}(\sqrt{-\Delta_g})$. The randomization improves the bound on $\|\Pi_\lambda^* V_\omega \Pi_\lambda\|$, replacing the deterministic $L^q$ dependence with a probabilistic $\ell^{2q}$ dependence on the coefficients, gaining a factor of $\lambda^{-\mu(q)}$.
   • Half-Resolvents: The authors derive deterministic estimates for the square root of the resolvent $|-\Delta_g - z|^{-1/2}$. These are combined with the randomized multiplier bounds to estimate the Birman-Schwinger operator $K_{\omega, \lambda}(z) = |-\Delta_g - z|^{-1/2} V_{\omega, \lambda} (-\Delta_g - z)^{-1/2}$.

3. Discretization and Union Bound:
   To extend the pointwise probabilistic bounds on the Birman-Schwinger operator to a uniform bound over the continuous spectral parameter $z$, the authors employ a discretization scheme. They partition the complex plane based on the magnitude of the resolvent weight $B(z)$, construct an $\epsilon$-net, and apply a union bound over the net points, utilizing the Lipschitz continuity of the Birman-Schwinger operator with respect to $z$.

Key Contributions and Results

• Theorem 1.1 (Spectral Inclusion): The main result provides a high-probability bound on the spectrum of $H_{\omega, \lambda}$. It states that with probability at least $1 - \exp(-K^2)$, any eigenvalue $z$ with $|z| \leq \lambda^2$ and distance to the unperturbed spectrum $\geq \lambda^{-N}$ lies within a union of disks centered at the Laplacian eigenvalues $\lambda_k^2$. The radius of these disks, $r_k$, and the exclusion region $\Omega$ depend on the $\ell^{2q}$-norm of the potential coefficients $v(\lambda)$, the dimension $d$, and the parameter $q$.

  • Square-Root Cancellation: The bound exhibits a gain over deterministic results (specifically those in [Cue26]). The deterministic bound scales with $\|V\|_{L^q}$, whereas the randomized bound scales with $\|v\|_{\ell^{2q}}$ with an improved power of $\lambda$. The authors note that for $q > (d+1)/4$, the probabilistic bound is strictly better than the deterministic one, effectively achieving a square-root cancellation in the potential's norm.

• Theorem 1.2 and 1.4 (Abstract Estimates): These theorems provide the technical machinery for the main result. They establish that for operators $T_1^* V_\omega T_2$, the expected operator norm is bounded by a term involving $(\log \#E)^{5/2}$ (discrete) or $(\log \#\Lambda)^{5/2}$ (continuous) and the $\ell^{2q}$ (or $L^{2q}$) norm of the coefficients, rather than the $L^\infty$ norm.

• Proposition 4.1 and 4.6: These intermediate results quantify the improvement in spectral cluster estimates and Birman-Schwinger bounds, respectively, explicitly showing the dependence on the randomization scale and the probabilistic gain.

Significance and Claims
The paper claims that its primary contribution is a "general abstract framework" that separates the probabilistic input (randomization of coefficients) from the analytic input (deterministic spectral bounds).

• Independence of Geometry: The abstract results are formulated for general Ahlfors regular metric measure spaces, suggesting applicability beyond the specific compact manifold setting.
• Improvement over Deterministic Bounds: The authors explicitly state that their results yield a "square-root cancellation gain" compared to deterministic spectral inclusion bounds. This is a significant improvement in the high-energy regime, where the deterministic bounds are known to be sharp (e.g., on spheres or Zoll manifolds).
• Broad Applicability: The authors identify the mechanism as applicable to various settings, including:
  • Manifolds with boundary and non-compact spaces (asymptotically conic or hyperbolic).
  • Random Schrödinger operators on $\mathbb{R}^d$ with trapping potentials or magnetic fields.
  • Discrete models on lattices ($\mathbb{Z}^d$) and graphs.
  • Questions regarding the existence and completeness of wave operators and the absence of embedded eigenvalues in random settings.

The paper concludes that while the specific bounds for compact manifolds are the primary focus, the underlying mechanism of randomization improving operator norm bounds is a robust principle with potential extensions to a wide class of spectral problems in mathematical physics.