The Pseudospectrum of Systems of Semiclassical Operators
Nils Dencker
备注：
Introduced the notion of quasi-symmetric systems and proved invariance properties for these systems. Changed the definition of the approximation property in Definition 5.10. Skipped erroneous Remark 3.5 and strengthened Proposition 3.7. Corrected an error in the proof of Proposition 6.1, added Lemma A.4 and some examples. The paper has 57 pages
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。
英文摘要：
The pseudospectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. In fact, for non-selfadjoint operators the resolvent could be very large outside the spectrum, making the numerical computation of the complex eigenvalues very hard. This has importance, for example, in quantum mechanics, random matrix theory and fluid dynamics.
The occurence of pseudospectra for non-selfadjoint semiclassical differential operators is due to the existence of quasimodes, i.e., approximate local solutions to the eigenvalue problem. For scalar operators, the quasimodes appear since the bracket condition is not satisfied for topological reasons, see the paper by Dencker, Sjostrand and Zworski in Comm. Pure Appl. Math. 57 (2004), 384-415.
In this paper we shall investigate how these result can be generalized to square systems of semiclassical differential operators of principal type. These are the systems whose principal symbol vanishes of first order on its kernel. We show that the resolvent blows up as in the scalar case, except in a nowhere dense set of degenerate values. We also define quasi-symmetrizable systems and systems of subelliptic type for which we prove estimates on the resolvent.
我们考虑一个非线性演化问题的渐近参数和构造的例子中，线性化的操作有谱一致有界的Re z >= 0（即，该问题是谱稳定的），但非线性演化爆破在很短的时间内为任意小的初始数据。我们用线性化算子的半经典伪谱来解释结果：尽管在Re z < -c < 0中有谱，线性化算子的预解式在Re z > 0的部分区域增长非常快。我们还说明了数值结果。
We obtain sharp uniform bounds on the low lying eigenfunctions for a class of semiclassical pseudodifferential operators with double characteristics and complex valued symbols, under the assumption that the quadratic approximations along the double characteristics are elliptic.
In this paper we prove a subelliptic resolvent estimate for a class of semiclassical non-self-adjoint Schrödinger operators with purely imaginary potentials when the spectral parameter is in a parabolic neighborhood of the imaginary axis.
We develop a set of $L^{p}$ estimates for functions $u$ that are a joint quasimodes (approximate eigenfunctions) of $r$ semiclassical pseudodifferential operators $p_{1}(x,hD),\dots,p_{r}(x,hD)$. This work extends Sarnak and Marshall's work on symmetric space to cover a more general class of manifolds/operators.
Spectral analysis of Dirac operators for dislocated potentials with a purely imaginary jump
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University（赫瑞-瓦特大学数学系和麦克斯韦数学科学研究所）；Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague（布拉格捷克技术大学核科学与物理工程学院数学系）；Faculty of Mathematics and Statistics, Ton Duc Thang University（胡志明市统一大学数学与统计学院）
In this paper we develop $L^{p}$ estimates for functions $u$ which are joint quasimodes of semiclassical pseudodifferential operators $p_{1}(x,hD)$ and $p_{2}(x,hD)$ whose characteristic sets meet with $k$th order contact, $k\geq{}1$. As part of the technical development we use Fourier integral operators to adapt a flat wavelet analysis to the curved level sets of $p_{1}(x,ξ)$.
We prove uniform $L^p$ resolvent estimates for the stationary damped wave operator. The uniform $L^p$ resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Dos Santos Ferreira-Kenig-Salo and advanced further by Bourgain-Shao-Sogge-Yao. Here we provide an alternative proof relying on the techniques of semiclassical Strichartz estimates. This approach allows us also to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semiclassical spectral analysis framework.
This paper establishes an aspect of Bohr's correspondence principle, i.e. that quantum mechanics converges in the high frequency limit to classical mechanics, for commuting semiclassical unitary operators. We prove, under minimal assumptions, that the semiclassical limit of the convex hulls of the quantum spectrum of a collection of commuting semiclassical unitary operators converges to the convex hull of the classical spectrum of the principal symbols of the operators.
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 obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration in [CCF16]. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.
We define a Schrödinger operator on the half-space with a discontinuous magnetic field having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we study the infimum of the spectrum of the operator. We give sufficient conditions on the strength and the direction of the magnetic field such that the aforementioned infimum is an eigenvalue of a reduced model operator on the half-plane. We use the Schrödinger operator on the half-space to study a new semiclassical problem in bounded domains of the space, considering a magnetic Neuma…
In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schr{ö}dinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifold M. We use this normal form to get an expansion of the first eigenvalues in powers of h^{1/2}, and semiclassical Weyl asymptotics for this operator.
We study semiclassical Gevrey pseudodifferential operators acting on the Bargmann space of entire functions with quadratic exponential weights. Using some ideas of the time frequency analysis, we show that such operators are uniformly bounded on a natural scale of exponentially weighted spaces of holomorphic functions, provided that the Gevrey index is $\geq 2$.
We study semiclassical Gevrey pseudodifferential operators, acting on exponentially weighted spaces of entire holomorphic functions. The symbols of such operators are Gevrey functions defined on suitable I-Lagrangian submanifolds of the complexified phase space, which are extended almost holomorphically in the same Gevrey class, or in some larger space, to complex neighborhoods of these submanifolds. Using almost holomorphic extensions, we obtain uniformly bounded realizations of such operators on a natural scale of exponentially weighted spaces of holomorphic functions for all Gevrey indices,…
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.
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 define classes of quantum states associated to isotropic submanifolds of cotangent bundles. The classes are stable under the action of semiclassical pseudo-differential operators and covariant under the action of semiclassical Fourier integral operators. We develop a semiclassical symbol calculus for them; the symbols are symplectic spinors. We outline various applications.
Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\mathbb{C}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.
We consider the operator ${\mathcal A}_h=-Δ+iV$ in the semi-classical $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.
