5 problems

Let $\Omega_x$ be a domain with $0<x<1$ and graph approximations $\{\Omega_{x,m}\}$. Suppose that the boundaries change when passing from the $m$-level graph approximation to the…

Let $\mathcal{S}_m$ be the level-$m$ spectrum. Accumulation conjecture below 5. For every $\varepsilon>0$, the number of distinct eigenvalues of $\mathcal{S}_m$ lying between…

Let $\mathcal{S}$ be the Dirichlet spectrum of $-\Delta$ on $\Omega$, and consider consecutive eigenvalues $\lambda<\lambda'$. Ratio-gap conjecture. There are infinitely many pairs…

Let $\rho^\Omega(x)$ be the eigenvalue counting function of the Dirichlet spectrum on $\Omega$, and let $c$ be the constant arising from the limiting largest eigenvalue in the fini…

Let $\mathcal{S}_m$ be the Dirichlet spectrum at level $m$, and let $\rho_m^\Omega(x)=\sharp\{\lambda_m\in\mathcal{S}_m:\lambda_m\leq x\}$ be its eigenvalue counting function. Let…