N. Burq, P. Gerard, N. Tzvetkov
The purpose of this note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained in a previous work in dimension 2. We also give some related trilinear estimates.
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We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold $M$. We prove global existence of strong $H^1$ solutions on $M=S^3$ and $M=S^2\times S^1$ as far as the nonlinearity is defocusing and sub-quintic and thus we extend the results of Ginibre-Velo and Bourgain who treated the cases of the Euclidean space $\R^3$ and the flat torus $\T^3$ respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.
We consider a Schrödinger operator with bounded, measurable potential in multidimensional Euclidean space. We prove for every $L^2$-eigenfunction a quantitative equidistribution estimate. It compares the total $L^2$-norm with the $L^2$-norm over an equidistributed collection of balls. Our estimate is explicit with respect to the radius of the balls, norm of the potential and the energy of the eigenfunction. Similar estimates also hold for Weyl sequences and for linear combinations of eigenfunctions, as long as the associated eigenvalues are sufficiently close.
We prove sharp bilinear estimates for Dirichlet or Neumann eigenfunctions in domains in the plane. These are the natural analog of earlier estimates for the boundaryless case of Burq, Gérard, and Tzvetkov.
Christ and Kiselev have established that the generalized eigenfunctions of one-dimensional Dirac operators with $L^p$ potential $F$ are bounded for almost all energies for $p < 2$. Roughly speaking, the proof involved writing these eigenfunctions as a multilinear series $\sum_n T_n(F, ..., F)$ and carefully bounding each term $T_n(F, ..., F)$. It is conjectured that the results of Christ and Kiselev also hold for $L^2$ potentials $F$. However in this note we show that the bilinear term $T_2(F,F)$ and the trilinear term $T_3(F,F,F)$ are badly behaved on $L^2$, which seems to indicate that multi…
We use the sum-of-squares theorem from number theory to construct eigenfunctions of the Laplacian on the $d$-dimensional torus, $d \geq 2$, which vanish to any prescribed order at some point. These functions are then applied to provide a negative answer (in dimension $d \geq 2$) to a question in the context of quantitative unique continuation for spectral projectors of Schrödinger operators, asked by Egidi and Veselić.
We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian, and to the boundedness of $L^p$ norms of eigenfunctions of the Laplacian. We formulate a conjecture, and partially prove it.
We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere $\mathbb{S}^2$ and more generally on Zoll surfaces, we prove that, for $s>1/4$, NLS is uniformly well-posed in $H^s$, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. On étudie l'équation de Schrödinger non linéaire (NLS) sur une surface compacte.Sur la sphère $\mathbb{S}^2$ et plus généralement sur toute surface de Zoll, on démontre que pour $s>1/4$, NLS est uniformément bien posée dans $H^s$, ce qui est opt…
Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as λ tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most λ. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector onto the frequency interval (λ, λ+ 1] has a universal scaling limit as λ goes to infinity (depending only on the dimension of M). Our results also imply that if M has no conjugate points, then immersions of M into Euclidean space by an orthono…
On the {$L^\infty $} norms of spectral projectors on shrinking intervals: the cases of some spheres of revolution and of the Euclidean disk
本文研究了旋转球面和欧几里得圆盘上谱投影的$ L^\infty $范数，利用量子可积性和渐近拉格朗日振荡函数，通过分析caustics分布和BKW衰减，实现了多项式改进的计数估计。
$L^4$ norm of spectral projectors on polynomially small frequency intervals for $S^1$-symmetric surfaces
研究紧致黎曼曲面上谱投影在多项式小带宽下的$L^2\to L^4$范数上界，通过Bessel本征函数分解和非平稳相位振荡积分估计，结合凸性现象和新的算术估计得到改进。
Quantum ergodicity for contact metric structures
本文在Reeb流遍历的假设下，利用半经典伪微分算子与Landau投影子，证明了接触度量流形上次拉普拉斯算子本征函数的量子遍历性定理。
Bounds for spectral projectors on the three-dimensional torus
本文研究三维环面上拉普拉斯算子谱投影子的$L^2$到$L^p$算子范数，在窄谱窗口情形下证明了猜想的新情形并消除$\epsilon$-损失，方法结合数论、高度分裂与双线性论证。
Sharp bilinear eigenfunction estimate, $L^\infty_{x_2}L^p_{t,x_1}$-type Strichartz estimate, and energy-critical NLS
本文通过利用球面的表示论，建立了三维球面上的精确双线性特征函数估计，并改进了各向异性Strichartz估计，从而获得了能量临界NLS在能量空间中的小数据全局适定性。
We consider the cubic non-linear Schrödinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$ follows easily from conservation of energy and standard arguments. In this work, we extend the range of global well-posedness to $s>2/3$. This generalizes, without any loss in regularity, a similar result on $\T^2$. The proof relies on the I-method of Colliander, Keel, Staffilani, Takaoka, and Tao, a semi-classical bilinear Strichartz estimate proved by the author, and s…
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem arises when studying localized perturbations around (possibly non-decaying) special solutions of evolution PDEs, and trying to control the projection onto the continuous spectrum of the nonlinear radiative interactions. One of our main tools is the Fourier transform adapted to the Schrödinger operator $H=-Δ+V$, which we employ at a nonlinear level. As a first st…
Sampling and equidistribution theorems for elliptic second order operators, lifting of eigenvalues, and applications
本文针对具有Lipschitz连续系数的椭圆二阶算子，证明了特征函数的尺度无关采样与等分布定理，并应用于特征值提升、本质谱下确界及谱不等式，克服了变系数带来的挑战。
Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds
本文借助多重线性特征函数估计、I-method及Strichartz估计，证明三维Zoll流形与球面积上低正则三次NLS的整体适定性，并给出$H^s$范数多项式界。
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