We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators in dimension $n \ge 1$. We require the potential belong to $L^\infty(\mathbb{R}^n)$ and have compact support, but do not require that it have derivatives in $L^\infty(\mathbb{R}^n)$. The weighted resolvent norm is bounded by $e^{Ch^{-4/3}\log(h^{-1})}$, where $h$ is the semiclassical parameter.
In this article, we shall construct the resolvent of Laplacian at high energies near the spectrum on non-product conic manifolds with a single cone tip. Microlocally, the resolvent kernel is the sum of b-pseudodifferential operators, scattering pseudodifferential operators and intersecting Legendrian distributions. As an application, we shall establish Strichartz estimates for Schrödinger equations on non-compact manifolds with multiple non-product conic singularities.
We give a uniform description of resolvents and complex powers of elliptic semiclassical cone differential operators as the semiclassical parameter $h$ tends to $0$. An example of such an operator is the shifted semiclassical Laplacian $h^2Δ_g+1$ on a manifold $(X, g)$ of dimension $n\geq 3$ with conic singularities. Our approach is constructive and based on techniques from geometric microlocal analysis: we construct the Schwartz kernels of resolvents and complex powers as conormal distributions on a suitable resolution of the space $[0,1)_h\times X\times X$ of $h$-dependent integral kernels; …
We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) - E$ in dimension $n \neq 2$, where $h, \, E > 0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit long range decay at infinity, and may grow like a sufficiently small negative power of $r$ as $r \to 0$. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C > 0$. This $E$-depe…
In this note, we consider the wave operator $\square_g$ in the case of globally hyperbolic, compactly supported perturbations of static spacetimes. We give an elementary proof of the essential self-adjointness of $\square_g$ and of uniform microlocal estimates for the resolvent in this setting. This provides a model for studying Lorentzian spectral zeta functions which is particularly simple, yet sufficiently general for locally deriving Einstein equations from a spectral Lagrangian.
We construct a parametrix of a resolvent of elliptic differential operators acting on half-densities on manifolds with ends. The construction is carried out by introducing suitable pseudodifferential operators compatible with the end structure. Our class of pseudodifferential operators and symbols is independent of the choice of Riemannian metric on the manifold and applicable to both of asymptotically conical and hyperbolic manifolds. As an application, we prove the essential self-adjointness of elliptic symmetric differential operators on manifolds.
We first prove semiclassical resolvent estimates for the Schr{ö}dinger operator in R d , d $\ge$ 3, with real-valued potentials which are H{ö}lder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in R d , d $\ge$ 2, and real-valued potentials which are H{ö}lder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be H{ö}lder, Lipschitz or just L $\infty$ .
Semiclassical estimates of the cut-off resolvent for trapping perturbations
本文研究了半经典“黑箱”算子$P$ .我们用在远离原点的环上截断的预解式的范数来估计在陷集附近截断的预解式的范数。对于具有$- h |ln h| < Im z < 0$的非物理表中的$z$，我们证明了这个估计在常数$h |Im z|^{-1} e^{C|Im z|/h}$下成立。我们还得到了类似的$P$的共振态的边界。这些结果保持没有任何假设的陷阱集，也没有任何假设的多重共振。
We prove semiclassical resolvent estimates for the Schr{ö}dinger operator in R d , d $\ge$ 3, with real-valued radial potentials V $\in$ L $\infty$ (R d). We show that if V (x) = O x --$δ$ with $δ$ > 4, then the resolvent bound is of the form exp Ch -- $δ$ $δ$--1 log(h --1) 1 $δ$--1 with some constant C > 0. If V (x) = O e -- C x $α$ with C, $α$ > 0, we get better resolvent bounds of the form exp Ch --1 log(h --1
In this note, we prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at infinity or to obey a radial $α$-Hölder continuity condition, $0\leq α\leq 1$, with sufficient decay of the local radial $C^α$ norm toward infinity. Note, however, that in the Hölder case, the potential need \emph{not} decay. If the dimension $n \ge 3$, the resolvent bound is of the form $\exp \left(C h^{-1 - \frac{1 - α}{3 + α}} [(1-α) \log(h^{-1})+c]\right)$, while …
In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2π\mathbb{Z})^2$ with Hölder dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, …
We prove semiclassical resolvent estimates for the Schr{ö}dinger operator in R d , d $\ge$ 3, with real-valued radial potentials V $\in$ L $\infty$ (R d). In particular, we show that if V (x) = O x --$δ$ with $δ$ > 2, then the resolvent bound is of the form e Ch --4/3 with some constant C > 0. We also get resolvent bounds when 1 < $δ$ $\le$ 2. For slowly decaying $α$-H{ö}lder potentials we get better resolvent bounds of the form e Ch --4/($α$+3) .
The Pseudospectrum of Systems of Semiclassical Operators
非自伴算子的伪谱（或谱不稳定性）是当前应用数学中的一个热点问题。事实上，对于非自伴算子，预解式在谱外可能非常大，使得复特征值的数值计算非常困难。这在例如量子力学、随机矩阵理论和流体动力学中具有重要性。非自伴半经典微分算子伪谱的出现是由于准模的存在，即，本征值问题的近似局部解。对于标量算子，由于拓扑原因不满足括号条件，所以出现准模，参见Dencker、Sjostrand和Zworski在Comm. Pure Appl.Math.57（2004），384-415中的论文。本文将研究如何将这些结果推广到主型半经典微分算子的平方系统。这些系统的一阶主象征在其核上为零。我们表明，在标量的情况下，除了在一个疏集退化值的预解爆破。我们还定义了拟对称系统和系统的次椭圆型，我们证明估计的预解式。
We give examples of semiclassical Schrödinger operators with exponentially large cutoff resolvent norms, even when the supports of the cutoff and potential are very far apart. The examples are radial, which allows us to analyze the resolvent kernel in detail using ordinary differential equation techniques. In particular, we identify a threshold spatial radius where the resolvent behavior changes. We apply these results to wave equations with radial wavespeed, identifying a corresponding threshold radius at which wave decay properties change.
We study the following fractional Schrödinger equation \begin{equation*}\label{eq0.1} \varepsilon^{2s}(-Δ)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} where $s\in(0,1)$. Under some conditions on $f(u)$, we show that the problem has a family of solutions concentrating at any finite given local minima of $V$ provided that $V\in C(\R^N,[0,+\infty))$. All decay rates of $V$ are admissible. Especially, $V$ can be compactly supported. Different from the local case $s=1$ or the case of single-peak solutions, the nonlocal effect of the operator $(-Δ)^s$ makes the peaks of the candidate s…
We prove explicit semiclassical resolvent estimates for an integrable potential on the real line. The proof is a comparatively easy case of the spherical energies method, which has been used to prove similar theorems in higher dimensions and in more complicated geometric situations. The novelty in our results lies in the weakness of the assumptions on the potential.
We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a semiclassical Schrödinger operator is supposed to be smooth and to decrease at infinity, the usual formulas relating scattering quantities and resonances still hold. The main ingredient for the proofs is a resolvent estimate of a new type, relating the resolvent of an operator with the resolvent of its cut-off counterpart.
