Eigenvalue distribution for variable-coefficient fractional discretizations

Prove the eigenvalue distribution of the nonsymmetric spatial discretization matrices \(\mathcal{D}_N(a)\overline{G}_{\alpha,N}\) in the variable-coefficient case, specifically whether the eigenvalues are asymptotically distributed according to the symbol \(a(x)f_\alpha(\theta)\).

Background

The paper analyzes the spatial matrix sequence $\{\mathcal{D}_N(a)\overline{G}_{\alpha,N}\}_N$, where $\overline{G}_{\alpha,N}$ is the nonsymmetric Toeplitz matrix generated by the shifted Grünwald discretization of the left-sided Riemann–Liouville fractional derivative and $\mathcal{D}_N(a)$ samples the variable diffusion coefficient $a(x)$. The authors establish its GLT and singular-value distributions with symbol $a(x)f_\alpha(\theta)$.

Because the matrices are nonsymmetric, the GLT result does not directly imply an eigenvalue distribution. Numerical evidence suggests that the eigenvalues closely follow samples of $a(x)f_\alpha(\theta)$, but the paper states that a complete theoretical proof for this variable-coefficient setting is unavailable. Establishing this result would provide a rigorous eigenvalue-level counterpart to the singular-value and GLT analysis and would clarify the spectral foundation of the proposed preconditioners.