On the Spectrum of Schrödinger Operators Interacting at Two Distinct Scales

Technical Summary: On the Spectrum of Schrödinger Operators Interacting at Two Distinct Scales

Problem Statement
This paper investigates the spectral properties of the Schrödinger operator $\Delta - W_\varepsilon$ acting on the space of radially symmetric square-integrable functions, $L^2_{\text{rad}}(\mathbb{R}^3)$. The potential $W_\varepsilon$ is a sum of two radially symmetric components interacting at distinct spatial scales:
$$W_\varepsilon(x) = V_0(|x|) + V_{1,\varepsilon}(|x|),$$
where $V_{1,\varepsilon}(|x|) = \varepsilon^2 V_1(\varepsilon|x|)$. As $\varepsilon \to 0$, the component $V_{1,\varepsilon}$ becomes increasingly delocalized (acting in the far field), while $V_0$ remains localized (acting in the near field). The primary objective is to quantify the number of positive, discrete eigenvalues of $\Delta - W_\varepsilon$ and determine how this count is constructed from the spectra of the individual operators $\Delta - V_0$ and $\Delta - V_1$.

The study assumes the potentials satisfy specific smoothness and decay conditions (Assumption A1) and that zero is not an endpoint resonance for the individual operators (Assumption A2). The work is motivated by the linear stability analysis of solitons in nonlinear Schrödinger equations with multi-scale potentials, as well as reaction-diffusion systems.

Methodology
The authors employ a framework combining dynamical systems techniques with a rigorous separation of scales argument. The core of the analysis involves transforming the eigenvalue problem into a boundary value problem for a dynamical system on a compactified phase space.

1. Compactification and Desingularization: The radial domain $r \in [0, \infty)$ is compactified using variables $\sigma = r/(1+r)$ for the inner region and $\tau = \rho/(1+\rho)$ (where $\rho = \varepsilon r$) for the outer region. This maps spatial infinity to a finite boundary ($\sigma=1$ or $\tau=1$). Singularities at the origin ($r=0$) and the essential spectrum endpoint ($\lambda=0$) are resolved via time rescaling ($s = r + \ln r$) and spherical blow-up procedures, respectively.
2. Angular Flow and Sturm-Liouville Theory: The linear eigenvalue equation is reduced to an angular system describing the evolution of the angle $\theta = \arctan(v/p)$ of the solution vector in phase space. Eigenvalues correspond to heteroclinic orbits connecting specific equilibria at the boundaries of the compactified domain.
3. Separation of Scales and Matching: The problem is split into an "inner" system (dominated by $V_0$) and an "outer" system (dominated by $V_1$). The authors define a threshold region between these scales where the solutions are matched.
4. Shadowing and Gluing: Using bootstrap estimates and invariant manifold theory, the authors prove that solutions to the full problem "shadow" the solutions of the reduced inner and outer model problems. A topological gluing argument is then used to count the total number of eigenvalue crossings (phase rotations) by summing the contributions from the inner and outer regimes.

Key Contributions and Results
The paper establishes three main theorems characterizing the spectrum of $\Delta - W_\varepsilon$ for sufficiently small $\varepsilon$:

• Spectral Additivity (Theorem 1): The total number of positive eigenvalues of $\Delta - W_\varepsilon$ is exactly the sum of the number of positive eigenvalues of $\Delta - V_0$ and $\Delta - V_1$. Specifically, if $m(V)$ denotes the count of positive eigenvalues for a potential $V$, then:
  $$m(W_\varepsilon) = m(V_0) + m(V_1).$$
  Furthermore, the operator has no endpoint resonance at $\lambda = 0$.
• Scale-Dependent Eigenvalues (Theorem 2 & Corollary 4): The eigenvalues are separated by order of magnitude:
  • There are exactly $m(V_1)$ eigenvalues of order $O(\varepsilon^2)$, arising from the far-field potential $V_{1,\varepsilon}$.
  • There are exactly $m(V_0)$ eigenvalues of order $O(1)$, arising from the near-field potential $V_0$, which are perturbations of the eigenvalues of $\Delta - V_0$.
• Continuity and Decay Analysis: The authors provide a detailed analysis of the behavior of the distinguished orbits at the endpoint of the essential spectrum ($\lambda=0$). They prove the continuity of these orbits with respect to the eigenvalue parameter and establish specific decay rates ($O(1/r^2)$) required to ensure the validity of the matching argument.

Significance and Claims
The paper claims to provide a novel framework for studying spectral properties of differential operators where multiple spatial scales interact. Its primary significance lies in:

• Rigorous Spectral Additivity: It demonstrates that for scale-separated potentials, the discrete spectrum is additive. This resolves the question of how the spectra of the individual components combine without interaction terms creating or destroying eigenvalues in the limit.
• Critical Scaling: The authors note that their scaling $V_{1,\varepsilon} = \varepsilon^2 V_1(\varepsilon r)$ is sharp with respect to the Cwikel-Lieb-Rozenbljum (CLR) bound. If the scaling were stronger ($r > 2$), the problem would be globally perturbative to $V_0$; if weaker ($r < 2$), it would be perturbative to $V_1$. The chosen scaling makes the problem critical, requiring the specific separation of scales analysis presented.
• Methodological Framework: The combination of dynamical systems (blow-up, invariant manifolds) with spectral counting offers a robust method for analyzing operators where standard perturbation theory might fail due to the distinct scales of the potentials.

The authors explicitly state that this work grew out of the need to understand scalar spectral considerations for the stability of solitons in nonlinear Schrödinger equations with multi-scale potentials, serving as a foundational step before addressing the more complex non-self-adjoint matrix systems found in those applications. The results are verified numerically across three distinct classes of potentials (Gaussian, rational, and hyperbolic), confirming the theoretical eigenvalue counts.