15 problems

Triangle eigenvalue bound. The eigenvalue satisfies

Let $D$ be a convex bounded domain of diameter $d_D$, and let $u1(D)< u2(D)$ denote the first two eigenvalues of its Dirichlet Laplacian. The spectral gap is $u2(D)- u1(D)$. van…

Let $\Omega\subseteq\mathbb{R}^2$ range over planar domains, and let $\lambda_1(\Omega)\leq\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\cdots$ be the Dirichlet Laplacian eigenvalues.…

Let $\Omega\subseteq\mathbb{R}^2$ be a connected planar domain, and let $u_3$ be an eigenfunction corresponding to a simple third eigenvalue $\lambda_3$ of the Dirichlet Laplacian…

Let $\Omega$ be a planar domain, and let $\lambda_1(\Omega)<\lambda_2(\Omega)\leq\lambda_3(\Omega)\leq\cdots$ denote the Dirichlet Laplacian eigenvalues. A local maximum is underst…

Let $\Omega$ be a bounded domain in an $n$-dimensional complete Riemannian manifold $M^n$, let $\lambda_k$ be the eigenvalues of the Dirichlet Laplacian on $\Omega$, and set…

Let $\Omega\subset\mathbb{R}^2$ be an open domain, and let $H_0^\Omega$ be its non-magnetic Dirichlet Laplacian. Write … for the eigenvalue counting function and let $|\Omega|$ den…

Let $\lambda_j$ denote the Dirichlet Laplacian eigenvalues of an open planar set of constant width, and consider the unresolved cases associated with the zeros $j_{7,1}$, correspon…

Let $\Omega$ be an open set in the Euclidean plane with constant width, and let $\lambda_3(\Omega)$ denote its third Dirichlet Laplacian eigenvalue. The disk is a candidate for min…

Let $\Omega\subset\mathbb{R}^2$ be a bounded domain with prescribed diameter, and let $\lambda_1(\Omega)\leq\lambda_2(\Omega)\leq\cdots$ denote the Dirichlet-Laplacian eigenvalues…

Let $\lambda_j$ be the eigenvalues of the Dirichlet Laplacian, listed with multiplicity, and let $k\in\mathbb N$. Consider linear combinations of eigenfunctions associated with eig…

Let $\Omega\subset\mathbb{R}^d$ be a domain with volume $|\Omega|$, let $\lambda_i$ be the eigenvalues of its Dirichlet Laplacian, and let $t>0$. Melas' heat-trace conjecture. The…

Let $\Omega\subset\mathbb{R}^d$ be a domain with volume $|\Omega|$, and let $\zeta_{\mathrm{spec}}(\rho)=\sum_{j=1}^{\infty}\lambda_j^{-\rho}$ denote the spectral zeta function of…

Let $\Omega\subset\mathbb{R}^d$ be a domain with volume $|\Omega|$, let $\lambda_j$ be the eigenvalues of its Dirichlet Laplacian, and let $F$ and $G$ be related by the transforms…

Let $\Omega_\varepsilon$ be the thin planar domain considered in the paper, let $-\Delta_{\Omega_\varepsilon}^D$ denote its Dirichlet Laplacian, and let $\lambda_{1,m}$ be the eige…