L^p norms of eigenfunctions in the completely integrable case

可积台球上高能本征函数的局部极限

Local limits of high energy eigenfunctions on integrable billiards

本文研究可积台球中高能本征函数的逆局域化性质，发现大多数可积多边形和椭圆的局部极限显著偏离Berry猜想预期的随机波。

Concentration of quantum integrable eigenfunctions on a convex surface of revolution

Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $φ^\ell_m$ be the orthonormal basis of joint eigenfunctions of $Δ_g$ and $\partial_θ$, the generator of the rotation action. The main result is an explicit formula for the weak-* limit of the normalized empirical measures, $Σ_{m = -\ell}^\ell ||φ^\ell_m||^2_{L^2(H)} δ_{\frac{m}{\ell}}(c)$ on $[-1,1]$. The explicit formula shows that, asymptotically, the $L^2$ norms of restricted eigenfunctions are minimal for the zonal eigenfunction $m = 0$, maximal for Gaussian beams $m = \pm 1$…

Pointwise bounds for joint eigenfunctions of quantum completely integrable systems

Let $(M,g)$ be a compact Riemannian manifold and $P_1:=-h^2Δ_g+V(x)-E_1$ so that $dp_1\neq 0$ on $p_1=0$. We assume that $P_1$ is quantum completely integrable in the sense that there exist functionally independent pseuodifferential operators $P_2,\dots P_n$ with $[P_i,P_j]=0$, $i,j=1,\dots ,n$. We study the pointwise bounds for the joint eigenfunctions, $u_h$ of the system $\{P_i\}_{i=1}^n$ with $P_1u_h=E_1u_h+o(1)$. We first give polynomial improvements over the standard Hörmander bounds for typical points in $M$. In two and three dimensions, these estimates agree with the Hardy exponent $h^…

正则双曲镶嵌没有 $\ell^2$ 特征函数

Regular hyperbolic tilings have no $\ell^2$ eigenfunctions

本文证明双曲平面正则镶嵌的1-骨架邻接算子无非零平方可积特征函数，并推广到具有正则对偶的无限连通正则图。

Symplectic Techniques for Semiclassical Completely Integrable Systems

This article is a survey of classical and quantum completely integrable systems from the viewpoint of local ``phase space'' analysis. It advocates the use of normal forms and shows how to get global information from glueing local pieces. Many crucial phenomena such as monodromy or eigenvalue concentration are shown to arise from the presence of non-degenerate critical points.

Riemannian Manifolds With Uniformly Bounded Eigenfunctions

The standard eigenfunctions $ϕ_λ = e^{i < λ, x >}$ on flat tori $\R^n / L$ have $L^{\infty}$-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that $L^2$-normalized eigenfunctions have uniformly bounded $L^{\infty}$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with completely integrable geodesic flows.