46 problems

For the relevant semiclassical magnetic Schrödinger inverse problems, the spectral data should determine the coefficients in the following symmetry-restricted settings: (i) an even…

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…

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 $r=1<r'$.…

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 $k$ 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…

Semiclassical transport conjecture. There exists $\beta<3/4$ such that, under the assumptions in, the solution $\Psi_t$ of

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 ${\cal P}:=T^*{\bf T}$ 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…

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…

Assume the conditions denoted by Assumptions. Let $\Pi_{\hbar,\mu}$ be the spectral projector defined in, and let … Let $g:[0,1]\to\mathbb{R}$ be continuous with $g(0)=g(1)=0$, and…

Low-frequency spectral asymptotics.

Let $(M,g)$ be a compact connected Riemannian manifold of negative sectional curvature. A weak limit $\tilde\mu$ is a weak- limit of the probability measures…

Let $\rho_Z$ be the one-body density of a ground state of the many-body atomic Hamiltonian with nuclear charge $Z$ and $N=N_c(Z)$ electrons. Universality conjecture. Along subseque…

Let $(M,g)$ be a compact manifold with Anosov geodesic flow, let $\mu$ be a semiclassical measure on $S^*M$, and let $\mu_L$ denote the Liouville measure. Liouville-component conje…

Confluent exponentiation conjecture for type $\mathcal G$. The confluent conformal block exponentiates semiclassically:

Let $(M,\omega)$ be a closed symplectic manifold. A geometric quantization is a sequence of maps $\{T_k:C^\infty(M)\to\mathcal{L}(H_k)\}_{k\in\mathbb{N}}$ satisfying the geometric…

Let $M$ be a connected manifold and let $V$ be a potential whose minima are non-degenerate in the sense of Shubin's Condition C. Let $\lambda_t$ denote the relevant analytic eigenv…

Let $\Omega$ be an ellipse with boundary $\Gamma$ of length $2L$, and let $s$ denote the curvilinear coordinate. Let $\lambda_1(h)$ and $\lambda_2(h)$ be the first two eigenvalues,…

Let $\rho_1,\rho_2$ be the ranks of the bulk Bloch bundles $\pazocal{E}_+$ and $\pazocal{E}_-$, and assume … Let $\pazocal{I}_e(P)$ denote the edge invariant, let $\mathbb{P}_h$ be…

Let $A^0$ be a scalar operator and let … be a strongly convex surface, meaning that … for all $\xi\in\Sigma^0_\lambda$ and all $\eta$ satisfying … where the sign depends on the con…

Semiclassical mean-field limit conjecture. For every function $f$ satisfying $f\geqslant 0$, $\sqrt{f}\in H^1(\mathbb{R}^d)$ and $\int_{\mathbb{R}^d}f=1$, one has

Suppose that $A$ and $S-S_\infty$ are Schwartz-class functions for some constant $S_\infty\in\mathbb{R}$, and that $k\in\mathbb{C}\setminus\{0\}$ admits a global classical solution…

Large-parameter positivity conjecture. There exists $y_1>0$ such that $\delta^T(y)>0$ on $[y_1,+\infty)$. The conjecture concerns the eventual monotonicity of the real part of the…

Consider pairwise commuting unitary semiclassical operators $U_1(\hbar), \ldots, U_d(\hbar)$ with joint principal symbol $F=(f_0^1,\ldots,f_0^d)$, and assume that $F(M)$ is closed.…