Sharp Ascent--Descent Spectral Stability under Strong Resolvent Convergence

TL;DR

This work addresses the stability of ascent and descent spectra for non-selfadjoint operators under strong resolvent convergence, a setting arising from finite element discretizations of convection–diffusion and Schrödinger-type problems. The authors identify the reduced minimum modulus $\gamma(S)$ as the key closed-range criterion and introduce a computable FEM surrogate $\gamma_h$ that predicts spectral stability and guides mesh refinement. They establish that stability holds under SRS if $\gamma((T-\lambda)^j)>0$ for all intermediate powers, with Kaashoek–Taylor criteria transferred via gap convergence; a Volterra counterexample shows the indispensability of considering all intermediate powers. The framework is validated through model operators and stabilized discretizations (notably SUPG) in 1D and 2D, demonstrating that $\gamma_h>0$ (uniformly) is necessary and sufficient for preserving ascent/descent indices in practice and explaining why norm-resolvent convergence can fail for rough limits. The results provide a practical, rigorous pathway to ensure spectral robustness in realistic computational settings, informing adaptive refinement and stabilization strategies for non-selfadjoint PDE discretizations.

Abstract

We establish sharp stability results for of non--selfadjoint the ascent and descent spectra under strong resolvent convergence (SRS), a natural framework for finite element approximations of non-selfadjoint and singularly perturbed operators. The key quantitative hypothesis is the reduced minimum modulus $γ(T-λ)>0$, which guarantees closed range and enables the transfer of the Kaashoek -- Taylor criteria via gap convergence of operator graphs. At the essential level, B--Fredholm theory extends stability to powers $(T-λ)^m$ provided $γ((T-λ)^j)>0$ for all $1\le j\le m$. We introduce a computable finite-element diagnostic $γ_h = σ_{\min}(M^{-1/2}(A_h-λM)M^{-1/2})$, which serves as a practical surrogate for $γ(T-λ)$ and remains uniformly positive even in convection-dominated regimes when stabilized schemes (e.g., SUPG) are employed. Numerical experiments confirm that $\liminf_{h\to0}γ_h>0$ is both necessary and sufficient for spectral stability, while a Volterra-type counterexample demonstrates the indispensability of the closed-range condition for powers. The analysis clarifies why norm resolvent convergence fails for rough or singular limits, and how SRS-combined with quantitative control of $γ_h$--rescues ascent--descent stability in realistic computational settings.

Figures (6)

• Figure 1: From numerical failure to abstract stability. The kernel/range chains stabilize at $\operatorname{asc}(S)$ and $\operatorname{dsc}(S)$. The Kaashoek–Taylor criteria link these to subspace transversality. Under strong resolvent convergence (SRS), graph convergence—and hence stability-is guaranteed precisely when $\gamma((T-\lambda)^j) > 0$ for all intermediate powers. The computable quantity $\gamma_h$ predicts success or failure in practice.
• Figure 2: Discrete reduced minimum modulus $\gamma_h$ versus mesh size $h$ for the Laplacian $-\partial_x^2$ on $(0,1)$ with Dirichlet boundary conditions and $\lambda = 25$. The eigenvalues satisfy $\zeta_1(h) \downarrow \pi^2 \approx 9.87$ and $\zeta_2(h) \downarrow 4\pi^2 \approx 39.48$ (Rayleigh--Ritz monotonicity, Proposition \ref{['prop:rayleigh-ritz']}). Since $\lambda \in (\pi^2, 4\pi^2)$, we have $\gamma_h = \min\bigl\{|25 - \zeta_1(h)|,\; |\zeta_2(h) - 25|\bigr\} \longrightarrow \min\{25 - \pi^2,\; 4\pi^2 - 25\} \approx 14.48.$ Numerical values are reported in Table \ref{['tab:laplacian']}.
• Figure 3: Discrete reduced minimum modulus $\gamma_h$ versus mesh size $h$ for convection--diffusion ($\varepsilon=0.02$, $\beta=8$, $c=0$) with $\lambda = -1$. The values remain uniformly bounded away from zero, confirming $\limsup_{h\to0}\gamma(S_h) > 0$. Numerical values are reported in Table \ref{['tab:cd-data']}, which also shows the lower bound $\operatorname{dist}(\lambda, W_M(A_h))$.
• Figure 4: (a) Discrete reduced minimum modulus $\gamma_h$ for the Schrödinger operator $-\partial_x^2 + V$ with $V \equiv 25$ on $(0,1)$, computed on a mesh with $N = 80$ interior nodes. Since the operator is selfadjoint, $\gamma_h(\lambda) = \operatorname{dist}(\lambda, \sigma(A_h,M))$; the lowest eigenvalue is $\zeta_1(h) \approx 34.87$. (b) $\gamma_h$ for the convection--diffusion operator with $\beta = 50$, $\lambda = -1$, and $h = 1/200$, as a function of $\varepsilon \in [10^{-3},1]$. The minimum value ($\approx 79.38$) confirms that $\gamma_h$ remains uniformly bounded away from zero even in the convection-dominated limit $\varepsilon \to 0$.
• Figure 5: Convergence rate of the discrete reduced minimum modulus $\gamma_h$ on the L-shaped domain. The reference slope $O(h)$ is shown for comparison. An extrapolated limit $\gamma_\infty \approx 2.60$ is used. The observed linear decay in the log--log scale confirms the theoretical expectation of $O(h)$ convergence, consistent with reduced elliptic regularity due to the reentrant corner.

Theorems & Definitions (53)

• Remark 3.1: Why SRS is natural in applications
• Theorem 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• proof : Proof of Theorem \ref{['thm:SRS']}
• Remark 3.5
• Remark 3.6
• Remark 3.7: On selecting the power level $m$ in practice