18 problems

Pseudospectral enclosure conjecture. There exists an $\varepsilon>0$ such that

Let $A$ be a matrix of arbitrary finite size, and consider the essential fault points associated with the boundary of its pseudospectrum. Finiteness conjecture. The number of essen…

Let $E$ be the operator appearing in the preceding conditions, with the first condition understood as the relevant $L^p$ boundedness condition. Isometricity conjecture. If $p=2$, t…

Gallay–Maekawa's conjecture. As $|\alpha|\to\infty$,

The paper considers a Hermitian matrix $A\in\mathbb{C}^{n\times n}$ perturbed by a diagonal random matrix whose entries are drawn from a floating-point distribution, so the distrib…

Pseudospectrum winding-number conjecture. The component of the $\varepsilon$-pseudospectrum containing $\lambda_0$ satisfies

Approximate joint-eigenstate conjecture. If $(\bm{\lambda},\nu)\in\bar{\Lambda}_{\epsilon}^{\mathrm{C}}(\bm{A},B)$, then there is a unit state $\bm{\psi}$ such that

Let $n$ be the matrix size and let $m\leq n$ be the model parameter, with fixed parameters $\delta$ and $a$. Spectral scaling conjecture. For both model 1 and model 2, when $n$ and…

Let $n$ be the matrix size, let $m\geq 1$ be fixed, and let $S_{\delta J}^{(2)}(m)$ denote model 2. Let $\lambda_0$ be the distinguished point included in its pseudospectrum, and l…

Let $n$ be the matrix size, let $m\geq 1$ be fixed, and let $S_{\delta J}^{(1)}(m)$ denote model 1. Let $\lambda_0$ be the distinguished point included in the pseudospectrum, and l…

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 $V_{\pm}\in\mathbb{C}$ with $\operatorname{Im}V_{+}\ne\operatorname{Im}V_{-}$, let $\alpha\in\mathbb{C}\setminus\{0\}$, and let $\mathscr{L}_{\alpha}$ be the corresponding Schr…

Let $A$ be a matrix, and let $\|A\|$ denote its operator norm. A matrix is spectrally stable when its eigenvector condition number $\kappa_V$ is bounded. Davies's conjecture. Every…

Let $H$ be a non-Hermitian Hamiltonian, and let its spectrum denote the usual spectral data while its pseudospectrum denotes the corresponding numerical construction used to assess…

Gallay's conjecture. There exists $C>0$ independent of $\alpha$ such that, as $|\alpha|\to+\infty$,

Pseudospectra conjecture. Michaelis–Menten enzyme kinetics and a whole host of other important models in biology also exhibit significant pseudospectra.

Let $A^b$ be a pseudo-ergodic random operator, let $\Sigma$ be its almost sure spectrum, and let $\pi_\infty$ and $\sigma_\infty$ denote the corresponding limiting pseudospectral a…

Boulton's conjecture. There exists $\varepsilon_0>0$ such that for all $0<\varepsilon<\varepsilon_0$, the corresponding $\varepsilon$-pseudospectrum satisfies the localization asse…