10 problems

Let $l,g$ be two even functions with $g>0$ on $(0,\pi)$, and suppose that $f\equiv l/g$ is monotone increasing over $(0,\pi)$. Set $X_n\equiv T_n^{-1}(g)T_n(l)$ for all $n$. Let th…

Let $\varepsilon$ denote the window parameter, let $\lambda_\varepsilon$ be the emerging eigenvalue, and let $c$ be a constant. Let the non-trivial solution of equation (1.9) satis…

Let $\textbackslash Sigma$ be an LQG domain, let $\textbackslash N(\textbackslash lambda)$ denote its eigenvalue counting function, and consider the next-order term beyond the lead…

Let $\textbackslash Sigma$ be a planar domain in the deterministic setting, let $N(\textbackslash lambda)$ be its Dirichlet eigenvalue counting function, and let…

Let $\textbackslash Sigma\textbackslash subset\textbackslash mathbb R^d$ be a domain with $d\textbackslash ge1$, and let $N(\textbackslash lambda)$ count the Dirichlet eigenvalues…

Let $l,g$ be two even functions with $g>0$ on $(0,\pi)$, and suppose that $f\equiv l/g$ is monotone increasing over $(0,\pi)$. Set $X_n\equiv T_n^{-1}(g)T_n(l)$ for all $n$. Let…

Let $l$ and $g$ be even, real-valued functions on $(-\pi,\pi)$, with $g$ nonnegative and not identically zero almost everywhere, and let $f=l/g$ be even and monotonic on $[0,\pi]$.…

Let $\{\tilde{\sigma}_j\}_{j\in\mathbb N_0}$ be the edge-wave and surface-wave quasi-eigenvalues arranged in ascending order. The sequence is asymptotically complete if the indexin…

Let $N(c3)$ be the eigenvalue counting function of the sloshing problem, and let $N^e(c3)$ and $N^s(c3)$ denote respectively the counting functions for edge-wave and surface-wave q…

Let $\lambda_0(r)$ denote the smallest generalized eigenvalue of the discretized self-adjoint problem, where $r$ is the thickness parameter. Power-law scaling conjecture. Based on…