155 problems

The Schrödinger operator $H=-\Delta+V$ on $\mathbb{R}^{d}$ has negative eigenvalues $\lambda_n(-\Delta+V)$, and the Lieb–Thirring inequality is … Here $V_-(x)=\max\{0,-V(x)\}$, and…

Quantum unique ergodicity conjecture. It is not necessary to pass to a density-one subsequence: the entire sequence converges to the Liouville measure in the sense of semiclassical…

Let $(\Sigma,g)$ be a closed hyperbolic surface, and consider high-frequency limits of Laplace eigenfunctions on it. A semiclassical defect measure is a measure describing such a h…

Let $\mathscr{N\!D}_n$ be the class of completely integrable systems on a $2n$-dimensional symplectic manifold with non-degenerate singularities. Let $Q(\mathscr{N\!D}_n)$ be the c…

Let $(M,g)$ be a Riemannian manifold, let $\varphi^t$ be its geodesic flow on $S^*M$, let $\mathcal M(\Delta_g)$ be the set of semiclassical measures of Laplace eigenfunctions, let…

The classical Hamiltonian dynamics is described by position $q$ and momentum $p$ satisfying … while the corresponding quantum Heisenberg operators $q_h$ and $p_h$ satisfy the analo…

Let $\mathbb{T}^n_\theta$ be the quantum torus, let $\Delta$ be its flat Laplacian, let $q>0$, set $r=2nq^{-1}$, and suppose that either $r\neq 1$ and $r'=\max(r,1)$, or $r=1<r'$.…

Zamolodchikov's conjecture. In the semi-classical limit, conformal blocks on a four-point sphere have an exponential structure.

Low-frequency spectral asymptotics.

Let $(M,g)$ be a hyperbolic surface, and let $u$ be a semiclassical measure arising as a weak-* limit of eigenfunctions on $S^M$. Colin de Verdière's conjecture. No semiclassical…

Let $M$ be a $2n$-dimensional Poisson manifold with a real polarization $P$ and two integrable systems given by Lagrangian fibrations $\pi_i:M\to B_i$, with generically transverse…

Let $f$ be an observable and consider eigenstates $\psi_j$ in the small energy window $\hbar k_j\in[1-\varepsilon,1+\varepsilon]$. Their quantum averages are…

Let $k$ be a positive integer, let $Q(\alpha,1)$ be the operator whose lowest eigenvalue is $\lambda_0(\alpha,1)$, and let $\hat{\nu}$ denote the minimum value of…

Convergence conjecture. In some suitable topology, each sequence $\{f_k^{[n]}\}_{n\geq 0}$ converges to $f$, for every $k=0,1,2,\dots$.

Let $\Delta^{\rm Poisson}(L)$ and $\Delta^{\rm RMT}(L)$ denote the Poisson and random-matrix predictions for the spectral rigidity, and let $q$ be the irregularity fraction of the…

Maximal singularity conjecture. No sequence of quasimodes, that is, images of the operators $\mathcal{\hat P}_{-T,T,\phi}$, can have a more singular limit measure than the singular…

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…

Additional shape-assumptions conjecture. Some additional assumptions on the shape of $A$ should be made.

Let $A$ be the operator under consideration, let $\Omega$ be its phase-space domain, and let $E_h$ and $E'_h$ denote the two transforms associated with $A$. Boundedness conjecture.…

The operator $H$ has eigenfunctions $\phi_n$ and eigenprojections $P_n$, and the preceding semiclassical approximation suggests an asymptotic estimate for $\lVert P_n\rVert$ as…

Existence conjecture. The equation

For $N$ anyons with oscillator confinement, let $\alpha_q$ be the dimensionless quantum statistical parameter. A semiclassical analysis produces eigenvalues that are linear in…

Consider $N$ anyons with oscillator confinement and statistical parameter $\alpha_q$. Their quantum spectrum contains eigenvalues that depend nonlinearly on $\alpha_q$; such eigenv…

Let ${\cal P}:=T^*{\bf T}$ be the classical phase space, with Hamiltonian $H(p,q):=\frac{1}{2}p^2+V(q)$ and Liouville measure $\lambda$. Let $\bar v:{\cal P}\to{\bf R}^d$ be the as…

Let $H(p,q):=\frac{1}{2}p^2+V(q)$ be the classical Hamiltonian on ${\cal P}:=T^*{\bf T}$, where ${\bf T}:={\bf R}^d/{\cal L}$ and $V\in C^\infty({\bf R}^d,{\bf R})$ is ${\cal L}$-p…