Herbert Koch, Fulvio Ricci
The twisted Laplacian in the d=2n dimensional Euclidean space has the spectrum n+2k, k a nonnegative integer. We find sharp asymptotic bounds of the norm of the projection to the eigenspace considered as map from L2 to Lp, for all p>2.
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In this note we obtain some Strichartz estimates for the Schrödinger equation associated to the twisted Laplacian on $\mathbb{C}^{n}\cong \mathbb{R}^{2n}$. The initial data will be considered in suitable Sobolev spaces associated to the twisted Laplacian.
Riesz transforms associated with the twisted Laplacian with drift
本文研究了与漂移扭曲拉普拉斯算子相关的任意阶里斯茨变换的强型和弱型有界性。
We study in this paper the global hypoellipticity property in the Gevrey category for the generalized twisted Laplacian on forms. Different from the 0-form case, where the twisted Laplacian is a scalar operator, this is a system of differential operators when acting on forms, each component operator being elliptic locally and degenerate globally. We obtain here the global hypoellipticity in anisotropic Gevrey space.
We establish the local well posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on $\C^n$ in certain first order Sobolev space. Our approach is based on Strichartz type estimates, and is valid for a general class of nonlinearities including power type. The case $n=1$ represents the magnetic Schrödinger equation in the plane with magnetic potential $A(z)=iz, z\in\C$.
Phase space analysis of spectral multipliers for the twisted Laplacian
本文利用元辛等价与转移原理，在调制空间和Wiener amalgam空间中研究扭曲拉普拉斯算子谱乘子的有界性，并完整刻画了其热流及分数阶热流的有界性。
Sublinear lower bounds of eigenvalues for twisted Laplacian on compact hyperbolic surfaces
该研究针对紧双曲曲面上与实调和1-形式相关的扭曲拉普拉斯算子，建立了其特征值数量的次线性下界，并证明量子唯一遍历性对某些扭曲拉普拉斯算子不成立。
We prove well-posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on $\C^n$ for a general class of nonlinearities including power type with subcritical case $0\leq α<\frac{2}{n-1}$, see Ratnakumar, Sohani (J. Funct. Anal. 2013). In this paper, we consider critical case $α=\frac{2}{n-1}$ with $n\geq 2$. Our approach is based on truncation of the given nonlinearity $G$, which is used by Cazenave Weissler (1989). We obtain solution for the truncated problem. We obtain solution to the original problem by passing to the limit.
On Hardy spaces associated with the twisted Laplacian and sharp estimates for the corresponding wave operator
本文刻画了扭曲 Laplace 算子对应的 Hardy 空间，并利用原子分解证明了相关波动算子从 Hardy 空间到 $L^p$ 的尖锐有界性条件。
Spectral multipliers III: Endpoint bounds, intertwining operators, and twisted Hardy spaces
该研究将自由拉普拉斯谱乘子的基本估计推广到带实标量势的受扰哈密顿量，得到Mihlin乘子精确界等结果，部分证实相关猜想并刻画扭曲Hardy空间。
We show that a twist of a three-dimensional tube of uniform cross-section yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in the tube. The proof employs Hardy inequalities for the Dirichlet Laplacian in twisted tubes and the method of self-similar variables and weighted Sobolev spaces for the heat equation.
We prove on-diagonal bounds for the heat kernel of the Dirichlet Laplacian $-Δ^D_Ω$ in locally twisted three-dimensional tubes $Ω$. In particular, we show that for any fixed $x$ the heat kernel decays for large times as $\mathrm{e}^{-E_1t}\, t^{-3/2}$, where $E_1$ is the fundamental eigenvalue of the Dirichlet Laplacian on the cross section of the tube. This shows that any, suitably regular, local twisting speeds up the decay of the heat kernel with respect to the case of straight (untwisted) tubes. Moreover, the above large time decay is valid for a wide class of subcritical operators defined…
We establish various Hardy-type inequalities for the Dirichlet Laplacian in perturbed periodically twisted tubes of non-circular cross-sections. We also state conjectures about the existence of such inequalities in more general regimes, which we support by heuristic and numerical arguments.
We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.
We study the spectrum of the Dirichlet Laplacian on an unbounded twisted tube with twisting velocity exploding to infinity. If the tube cross section does not intersect the axis of rotation, then its spectrum is purely discrete under some additional conditions on the twisting velocity (D.Krejcirik, 2015). In the current work we prove a Berezin type upper bound for the eigenvalue moments.
We consider the Dirichlet Laplacian $H_γ$ on a 3D twisted waveguide with random Anderson-type twisting $γ$. We introduce the integrated density of states $N_γ$ for the operator $H_γ$, and investigate the Lifshits tails of $N_γ$, i.e. the asymptotic behavior of $N_γ(E)$ as $E \downarrow \inf {\rm supp}\, dN_γ$. In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.
In this paper we establish new quantitative stability estimates with respect to domain perturbations for all the eigenvalues of both the Neumann and the Dirichlet Laplacian. Our main results follow from an abstract lemma stating that it is actually sufficient to provide an estimate on suitable projection operators. Whereas this lemma could be applied under different regularity assumptions on the domain, here we use it to estimate the spectrum in Lipschitz and in so-called Reifenberg-flat domains. Our argument also relies on suitable extension techniques and on an estimate on the decay of the e…
The Conformal Fractional--Logarithmic Laplacian on the Sphere: Yamabe Problems and Sharp Inequalities
本文研究了单位球面上的共形分数-对数拉普拉斯算子的分析与谱性质，推导了Yamabe型方程，并应用于尖锐Sobolev型不等式，恢复了尖锐对数Sobolev不等式，揭示了分数-对数类比的失效，并建立了新的尖锐分数-对数不等式。
In this paper we consider the PDE system of vanishing normal projection of the Laplacian for $C^2$ maps $u : \mathbb{R}^n \supseteq Ω\longrightarrow \mathbb{R}^N$: \[ [\![\mathrm{D} u]\!]^\bot Δu = 0 \ \, \text{ in }Ω. \] This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the $p$-Laplace system for all $p\in [2,\infty]$. For $p=\infty$, the $\infty$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calc…
The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces
该论文研究球面上的共形对数拉普拉斯算子，建立其与ℝ^N中对数拉普拉斯算子的联系，引入对应希尔伯特空间，衔接Yamabe型问题分类结果并推广至弱情形。
We study $L^p$ bounds on spectral projections for the Laplace operator on compact Riemannian manifolds, restricted to small frequency dependent neighborhoods of submanifolds. In particular, if $λ$ is a frequency and the size of the neigborhood is $\mathcal{O}(λ^{-δ})$, then new sharp estimates are established when $δ\ge 1$, while for $0\le δ\le 1/2$, Sogge's estimates turn out to be optimal. In the intermediate region $1/2<δ<1$, we sometimes get sharp estimates as well. Our arguments follow closely a recent work by Burq and Zuily.
In this paper we consider certain asymptotically Euclidean spaces, namely compact manifolds with boundary X equipped with a scattering metric g, as defined by Melrose. We then consider Hamiltonians H which are `short-range' self-adjoint perturbations of the Laplacian of g. Melrose and Zworski have given a detailed description of the associated scattering matrix and Poisson operator as a Fourier integral operator and a (singular) Legendre distribution respectively. In this paper we describe the kernel of the spectral projections and the boundary value of the resolvent at the real axis. We defin…
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