A note on the convergence of the eigenvalues in a subdomain to the continuous spectrum

Technical Summary: Convergence of Eigenvalues in a Subdomain to the Continuous Spectrum

Problem Statement
This paper addresses a gap in the spectral analysis of the preconditioned operator $L^{-1}A$, defined as $\Delta^{-1}[\nabla \cdot (K\nabla u)]$, on a bounded, open, two-dimensional Lipschitz domain $\Omega$. Here, $K(x,y)$ is a continuous, symmetric, matrix-valued function, and the operator acts within a Sobolev space setting ($H^1_0(\Omega)$).

Previous work, specifically the SIGEST paper [4] and its predecessor [2], characterized the spectrum of this operator. A key result in [4] (Lemma 3.2) stated that if $K$ is a diagonal tensor constant on an open subdomain $S \subset \Omega$, the closed interval defined by the diagonal elements of $K$ within $S$ belongs to the spectrum of the preconditioned operator. While the statement of this lemma is correct, the paper identifies that the original proof contained a critical flaw. The previous argument attempted to construct global eigenfunctions by taking local solutions to a wave equation on a subdomain $\Sigma_l \subset S$ and extending them by zero to the entire domain $\Omega$. The authors demonstrate that this zero-extension fails to satisfy the weak formulation of the global generalized eigenvalue problem because the resulting boundary integrals over $\partial \Sigma_l$ do not vanish for arbitrary test functions. Consequently, local eigenfunctions cannot be directly "lifted" to global eigenfunctions.

Methodology
To rectify the proof and establish the spectral inclusion rigorously, the authors construct a Weyl singular sequence of approximate eigenfunctions. This approach avoids the need for exact global eigenfunctions, which do not exist for points in the continuous spectrum of a self-adjoint operator in a separable Hilbert space.

The methodology proceeds as follows:

1. Local Wave Equation: For a fixed $\lambda$ in the open interval $(k_1, k_2)$ (where $k_1, k_2$ are the constant diagonal values of $K$ in $S$), the generalized eigenvalue problem reduces locally to a wave equation $u_{yy} = c^2 u_{xx}$ with $c^2 = (\lambda - k_1)/(k_2 - \lambda)$.
2. Sequence Construction: Instead of a fixed subdomain, the authors define a sequence of rectangular subdomains $\Sigma_n$ centered at an arbitrary point $(x_0, y_0) \in S$, where the dimensions of $\Sigma_n$ shrink to zero as $n \to \infty$.
3. Approximate Eigenfunctions: The sequence of functions $u_n(x,y)$ is constructed as products of oscillatory sine modes modulated by a smooth cutoff function $f_n$. These functions are supported strictly within $\Sigma_n$ and vanish outside. The construction ensures that the functions satisfy the local wave equation approximately while maintaining specific oscillatory properties.
4. Verification of Weyl Sequence Properties: The authors prove two critical properties for the sequence $\{u_n\}$:
   • Non-vanishing Norm: The $H^1_0$ semi-norm of $u_n$ remains bounded away from zero as $n \to \infty$. This is achieved through careful scaling and the use of the Riemann-Lebesgue lemma to evaluate the asymptotic behavior of the integrals.
   • Residual Convergence: The residual of the generalized eigenvalue problem, $\|(A - \lambda L)u_n\|_{H^{-1}(\Omega)}$, converges to zero. This is demonstrated by splitting the residual integral into sine and cosine components ($I_n^{sin}$ and $I_n^{cos}$) and showing that both vanish in the limit through integration by parts and asymptotic analysis.

Key Results
The paper provides a rigorous proof of Lemma 3.2 from [4], establishing that the closed interval $[\sup \kappa_1, \inf \kappa_2]$ (assuming $\kappa_1 < \kappa_2$) is contained within the spectrum of the preconditioned operator $L^{-1}A$.

The analysis confirms that for any $\lambda$ in the open interval $(k_1, k_2)$, the operator does not possess a bounded inverse. The proof relies on the fact that if $\lambda$ were not in the spectrum, the inverse $(A - \lambda L)^{-1}$ would be bounded, leading to a contradiction where the norm of the sequence $u_n$ would have to vanish, violating the first property of the Weyl sequence.

Significance and Claims
The paper claims to refine the theoretical foundation of the spectral analysis presented in [4] and [2]. Its primary contribution is the correction of the proof mechanism regarding how local spectral properties relate to the global spectrum.

The authors highlight a specific theoretical insight: the transition from the local to the global perspective.

• On a fixed, shrinking subdomain $\Sigma_l$, the generalized eigenvalue problem admits a countably infinite set of eigenfunctions for a fixed $\lambda$ (due to the indefinite nature of the operator).
• As the subdomain shrinks to a point, these local eigenvalues do not converge to discrete eigenvalues of the global operator. Instead, they transform into points of the continuous spectrum.

This distinction is crucial because the preconditioned operator is self-adjoint on a separable Hilbert space and thus can have at most a countable set of eigenvalues. Since the spectrum in question is an interval, it cannot consist solely of eigenvalues. The paper clarifies that the interval is part of the continuous spectrum, arising from the non-compactness of the operator and the specific oscillatory behavior of the approximate eigenfunctions.

The authors conclude that while the spectral interval is confirmed to be part of the spectrum, it remains an open question whether this interval contains any actual eigenvalues of the global operator, though the construction suggests the spectrum is purely continuous in this region. The work underscores the necessity of using Weyl sequences rather than zero-extended local solutions when analyzing the spectrum of non-compact operators arising in the numerical treatment of elliptic PDEs.