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

The SIGEST paper Nielsen and Strako\v{s} (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 Strako\v{s} (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.