Model 2 conjecture on eigenvalue and pseudospectrum structures

Let $S_{\delta J}^{(2)}(m)$ denote the model 2 matrix, with matrix size $n$, parameters $\delta$ and $a$, and let $\lambda_0$ be the distinguished eigenvalue of the original system. Let $f_{\widehat{S}^m+a\widehat{S}^{m+1}}(\mathbb{T})$ be the symbol curve, whose outermost closed simple curve and inner part are defined by the observed decomposition of that curve. Model 2 eigenvalue and pseudospectrum conjecture. For model 2, at each time $m$, the dots composing the outermost curve in the numerical result are exact eigenvalues of $S_{\delta J}^{(2)}(m)$. For $m\geq 2$, the inner parts are not exact eigenvalues; they are eigenvalues of the system after uncontrolled computer-rounding perturbations are added. Their distributions represent pseudospectrum containing $\lambda_0$ and reflect the inner parts of the spectrum of the Toeplitz operator $\widehat{S}^m+a\widehat{S}^{m-1}$, without the deterministic perturbation by $\delta J$ or uncontrolled rounding perturbations. The size of this pseudospectrum decreases exponentially as $m$ increases. This conjecture proposes an explanation of numerically observed inner eigenvalue patterns as pseudospectral effects of rounding errors, while identifying the outermost eigenvalues as exact and robust spectral data.