81 problems

Modified Weyl–Berry conjecture. There is a positive constant $c_D$ depending only on $D$ such that

Let $\Omega\subset\mathbb R^n$ have a fractal boundary $\partial\Omega$ with Hausdorff dimension $H\in[n-1,n]$. Let ${\Greekmath 011A}(x)$ be the eigenvalue counting function for t…

Let the magnetic Dirichlet-to-Neumann operator be the operator associated with a magnetic field of strength tending to infinity, and let its ground state energy denote its lowest e…

Low-frequency spectral asymptotics.

Let $M^n$ be a closed $n$-dimensional manifold with $n\geq 2$, and let $\omega_p^1(M^n,g)$ denote its $1$-dimensional $p$-width with respect to a Riemannian metric $g$. The volume…

Ivrii's conjecture. The main part of the spectral asymptotics is given by ${\mathcal E}^{\rm {MW}}$, while: (i) for fixed $(g^{jk})$ and general $(V_j),V$, the remainder is…

Let ${\mathcal F}$ be an arbitrary foliation on a compact Riemannian manifold. Let $D_F=d_F+d^*_F$ be the leafwise signature operator on $M$, let $D_{{\mathcal F}_N}$ be the corres…

Let ${\mathcal F}$ be an arbitrary foliation on a compact Riemannian manifold. In the above notation, let $\Delta_F$ and $\Delta_H$ denote the leafwise and transverse Laplacians, r…

Arnol'd's conjecture. Studying the asymptotic behaviour of $N_\chi(\lambda)$ should show that the multiplicity of each unitary irreducible representation in the decomposition of…

Let $H(V)$ be the ergodic random Schrödinger operator considered in the paper, with vector potential $a=0$, and let $N(E)$ denote its integrated density of states. The density of s…

Let $\Gamma$ be a curve with free ends, and let $H_{\alpha,\Gamma}(h)$ be the associated singular-interaction Schrödinger operator in the semiclassical regime $h\to0$. Let $\mu_j$…

Consider two waveguides of widths $d_1$ and $d_2$, let … and define the waveguide-asymmetry parameter … Let $E(\ell)$ denote the eigenvalue associated with a narrow window of width…

On $\mathbb{T}^2$, let $\Delta_H$ be the Baouendi–Grushin sub-Laplacian and let $c>0$ be the constant in the singular Weyl measure $c\delta_S$, where $S=\{0\}\times\mathbb{T}$. Let…

On $\mathbb{T}^2$, let $X=\partial_x$, $Y=2(1-\cos x)\partial_y$, and let $\Delta_H=X^2+Y^2$. The singular set is $S=\{0\}\times\mathbb{T}$, and $c>0$ is the constant in the singul…

Let $M$ be a filtered manifold and let $\Psi_H$DOs denote its Heisenberg pseudodifferential operators. For a negative-order operator in this calculus, consider its singular values…

Let $(\mathscr{A},\mathscr{H},D)$ be a spectral triple with summability parameter $p$, and let $N(h^2D^2+V;\lambda)$ denote the relevant eigenvalue counting function. McDonald–Suko…

Let a curvilinear polygon have $n$ vertices with interior angles $\alpha_1,\dots,\alpha_n$. For $\alpha\in(0,2\pi)$, define … and … Form the multiset…

Let $\beta(\gamma,d)$ be the critical Robin parameter from Theorem, and let $\beta_W(\gamma,d-1)$ be the point where the second term in the two-term Weyl formula changes sign. Stri…

Let $\beta(\gamma,d)$ be the critical Robin parameter from Theorem, and let $\beta_W(\gamma,d-1)$ be the value at which the second term in the two-term Weyl formula changes sign. W…

Let $\Omega\subset\mathbb R^d$ be a domain, let $\beta$ be a Robin boundary parameter, and let $\gamma$ be the Riesz-mean order. Theorem gives a two-term semiclassical asymptotic e…

Let $(M,H)$ be a compact Carnot manifold, and let $T\in \Psi^{-d_H}_H(M)$ be a pseudodifferential operator on $(M,H)$ in the van Erp-Yuncken sense. Let $\nu$ be a density on $M$, i…

Let the integrated density of states (IDS) be the spectral distribution measure associated with a Schrödinger operator having a random potential. Lifshitz's conjecture. The IDS is…

Let $SG$ be the Sierpiński gasket, let $L$ be its bottom line, and let ${\Greekmath 011A}_D^{SG\setminus L}(x)$ denote the Dirichlet eigenvalue counting function on $SG\setminus L$…

Let $\Omega\subset\mathbb R^d$ be a domain of finite measure with inradius $\varrho$, and let $\lambda_{1}^{(m)}(\Omega)$ denote its first polyharmonic eigenvalue. Inradius asympto…

Universality conjecture. For every such $\Omega$,