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

The SIGEST paper Nielsen and Strakoš (2024) characterized the spectrum of the preconditioned operator $\Delta^{-1}[\nabla \cdot (K\nabla u)]$ in a bounded open two-dimensional domain $\Omega$, where $\Delta$ denotes the Laplacian and $K(x,y)$ is a continuous symmetric matrix-valued function. An important part of the analysis states that for a diagonal tensor $K$ constant in an open subdomain $S \subset \Omega$, the closed interval defined by its diagonal elements belongs to the spectrum of the preconditioned operator. This result is correct, but the proof in Nielsen and Strakoš (2024) must be refined.
This paper presents a refined proof and extends the previous work. As shown in the cited papers, for any point $\lambda$ in the open interval defined by the elements of the diagonal tensor constant in $S$ and any point $(x_0,y_0)\in S$, a rectangular subdomain $\Sigma_l\subset S$ can be constructed such that the generalized eigenvalue problem associated with the preconditioned operator restricted to $\Sigma_l$, of arbitrarily small size, has the eigenvalue $\lambda$ and infinitely many eigenfunctions. These are given by solutions of a locally defined wave equation. However, such solutions of the locally restricted generalized eigenvalue problem cannot be extended to the whole domain $\Omega$. Using instead rectangular subdomains whose size shrinks to zero, the present paper constructs a Weyl singular sequence of \emph{approximate} eigenfunctions associated with $\lambda$, proving that $\lambda$ belongs to the spectrum of the preconditioned operator. Since self-adjoint operators in a separable Hilbert space can have at most a countable set of eigenvalues, this shows that the eigenvalues of the locally defined operator converge to points of the continuous spectrum of the preconditioned operator on the entire domain.