19 problems

Let … where $g$ is real, so that the cubic oscillator has a purely imaginary coupling constant. Bessis's reality conjecture. All eigenvalues of $H$ are real. The conjecture concern…

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

For each randomly distributed $N\times N$ matrix, let $P_n$ denote its spectral projections and consider the logarithms of their norms, sorted increasingly. Form the covariance mat…

For randomly generated tridiagonal $N\times N$ matrices, let ${\cal P}_{50}$ denote the value below which $50\%$ of the instability indices lie, where the instability index is the…

Let $L=-\epsilon\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)-\frac{\partial}{\partial\theta}$ act on sufficiently smooth periodic functions…

Let $H_c$ be the complex Schrödinger operator considered above, and consider curves of the form … where $b>0$, $\eta>0$, and $p$ is independent of $\eta$. Boulton's critical-index…

Let $P$ be an elliptic positive polynomial on $\mathbb{R}^n$ of degree $M\geq 2$, and consider the quadratic operator family … on $\mathcal{L}^2(\mathbb{R}^n)$. Helffer–Robert–Wang…

Consider the adjoint-free operator learning algorithm applied to second-order uniformly elliptic partial differential equations perturbed away from self-adjointness by lower-order…

Let $H$ be the non-self-adjoint operator considered in the paper, and suppose that $H$ has a spectral singularity. Let the regularized wave operators be the wave operators obtained…

Let $\mathfrak{D}_{\hbar}$ be the non-self-adjoint Dirac operator associated with a general multi-humped decaying potential, and consider its eigenvalues in the semiclassical limit…

Let $A$ satisfy the multi-hump potential assumption, and let $obreakmathfrak{D}{hbar}$ be the associated non-self-adjoint Dirac operator. Write $sigmap(mathfrak{D}{hbar})$ for its…

Hansmann–Katriel conjecture. The inequality above is no longer true when the assumption that $J$ is self-adjoint is dropped. The conjecture concerns the extension of Lieb–Thirring…

Let … and let the parity operator $\mathcal{P}$ and complex conjugation operator $\mathcal{T}$ act by … The Hamiltonian is $\mathcal{PT}$-symmetric when…

A generalized banded Toeplitz matrix sequence is obtained by sampling coefficient functions along its diagonals; in the non-self-adjoint case, its eigenvalues need not be real. Bou…

Conjecture on unbroken $\mathcal{PT}$ symmetry. The eigenstates of a $\mathcal{PT}$-symmetric Hamiltonian $H$ with unbroken $\mathcal{PT}$ symmetry form a first type sequence.

Let $W=\Phi V$ satisfy Assumption (A1), where $\Phi\in\mathbb{C}\setminus\mathbb{R}e^{ik\frac{\pi}{2}}$ for some $k\in\mathbb{Z}$, and let $V:\mathbb{R}^3\longrightarrow\mathbb{R}$…

Spectral-bound conjecture. For sufficiently small $\epsilon>0$,

Consider the singular differential equation on $(-\pi,\pi)$ … where $0<\tilde\varepsilon<2$, with the periodic boundary conditions implicit in the problem. Reality conjecture. Desp…

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