Microlocal Analysis Research Articles

Equivariant fixed point formulae and Toeplitz operators under Hamiltonian torus actions and remarks on equivariant asymptotic expansions

Suppose given a holomorphic and Hamiltonian action of a compact torus on a polarized Hodge manifold. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of the torus on the associated Hardy space which decomposes into isotypes. The main result of this paper is the description of asymptotics along rays in weight space of traces of equivariant Toeplitz operators composed with quantomorphisms for the torus action. The main ingredient in the proof is the micro-local analysis of the equivariant Szegő kernels. As a particular case we obtain a simple approach for asymptotics of the Lefschetz fixed point formula and traces of Toeplitz operators in the setting of ladder representations. We also consider equivariant asymptotic when the decomposition given by the standard circle action is taken into account, in this case one can recall previous results of X. Ma and W. Zhang or of R. Paoletti. We address some explicit computations for the action of the special unitary group of dimension two.

Microlocal analysis of borehole seismic data

<p style='text-indent:20px;'>Borehole seismic data is obtained by receivers located in a well, with sources located on the surface or another well. Using microlocal analysis, we study possible approximate reconstruction, via linearized, filtered backprojection, of an isotropic sound speed in the subsurface for three types of data sets. The sources may form a dense array on the surface, or be located along a line on the surface (walkaway geometry) or in another borehole (crosswell). We show that for the dense array, reconstruction is feasible, with no artifacts in the absence of caustics in the background ray geometry, and mild artifacts in the presence of fold caustics in a sense that we define. In contrast, the walkaway and crosswell data sets both give rise to strong, nonremovable artifacts.</p>

Towards semi-classical analysis for sub-elliptic operators

We discuss the recent developments of semi-classical and micro-local analysis in the context of nilpotent Lie groups and for sub-elliptic operators. In particular, we give an overview of pseudo-differential calculi recently defined on nilpotent Lie groups as well as of the notion of quantum limits in the Euclidean and nilpotent cases.

Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

<p style='text-indent:20px;'>Boundary feedback stabilization of a <i>critical</i> third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word <i>critical</i> here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [<xref ref-type="bibr" rid="b37">37</xref>]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are invariant under the dynamics. The assumed smallness of the initial data is required only at the lowest energy level topology, which is sufficient to construct sufficiently smooth solutions to the nonlinear model.</p>

On the plate equation with nonstationary stochastic process coefficients

AbstractIn mechanics, while investigating the vibration of plates, we are frequently confronted with random perturbations from both internal and external environments, such as inhomogeneous materials, unstable temperature fluctuation, etc. In this respect, the plate equation with stochastic coefficients is used for modeling the randomness from various perturbations. Current paper is concerned with the regularity behavior of the solution of the stochastic plate equation, in particular, the joint influence from the degenerating and oscillating coefficients on the principal operator is of prime consideration. We apply the sophisticated techniques from microlocal analysis and stochastic analysis to explore the upper bound of loss of regularity. Furthermore, in order to demonstrate the optimality of the estimates, appropriate counter‐examples with periodic coefficients are constructed to show the lower bound of loss of regularity by the application of harmonic analysis and instability arguments.

Arthur packets for G2 and perverse sheaves on cubics

This paper begins the project of defining Arthur packets of all unipotent representations for the p-adic exceptional group G2. Here we treat the most interesting case by defining and computing Arthur packets with component group S3. We also show that the distributions attached to these packets are stable, subject to a hypothesis. This is done using a self-contained microlocal analysis of simple equivariant perverse sheaves on the moduli space of homogeneous cubics in two variables. We treat all other unipotent representations of p-adic G2 in forthcoming work.

Detection of Hermitian connections in wave equations with cubic non-linearity

We consider the geometric non-linear inverse problem of recovering a Hermitian connection A from the source-to-solution map of the cubic wave equation \Box_{A}\phi+\kappa |\phi|^{2}\phi=f , where \kappa\neq 0 and \Box_{A} is the connection wave operator in the Minkowski space \mathbb{R}^{1+3} . The equation arises naturally when considering the Yang–Mills–Higgs equations with Mexican hat type potentials. Our proof exploits the microlocal analysis of non-linear wave interactions, but instead of employing information contained in the geometry of the wave front sets as in previous literature, we study the principal symbols of waves generated by suitable interactions. Moreover, our approach relies on inversion of a novel non-abelian broken light ray transform, a result interesting in its own right.

A microlocal approach to renormalization in stochastic PDEs

We present a novel framework for the study of a large class of nonlinear stochastic partial differential equations (PDEs), which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functional-valued distributions, we are able to use techniques proper of microlocal analysis which allow us to discuss renormalization and its associated freedom without resorting to any regularization scheme and to the subtraction of infinities. As an example of the effectiveness of the approach we apply it to the perturbative analysis of the stochastic [Formula: see text] model.

Limiting Absorption Principle and Radiation Condition for Repulsive Hamiltonians

For spherically symmetric repulsive Hamiltonians we prove the Besov bound, the radiation condition bounds and the limiting absorption principle. The Sommerfeld uniqueness result also follows as a corollary of these. In particular, the Hamiltonians considered in this paper cover the case of inverted harmonic oscillator. In the proofs of our theorems, we mainly use a commutator argument invented recently by Ito and Skibsted. This argument is simple and elementary, and dose not employ energy cut-offs or the microlocal analysis.

Ill-posedness issue for a multidimensional hyperbolic-parabolic model of chemotaxis in critical Besov spaces [formula omitted

In this paper, by fully exploiting nonlinear structure of the cross term in the chemotaxis system and using microlocal analysis, we prove that this system is ill-posed for the critical case B˙2d,1−32×(B˙2d,1−12)d in the sense of “norm inflation”, which is an unsolved problem in [21]. This result improves the ill-posedness theories of this system in critical Besov spaces framework.

Well-posedness for a modified nonlinear Schrödinger equation modeling the formation of rogue waves

The Cauchy problem for a higher order modification of the nonlinear Schrödinger equation (MNLS) on the line is shown to be well-posed in Sobolev spaces with exponent \(s > \frac{1}{4}\). This result is achieved by demonstrating that the associated integral operator is a contraction on a Bourgain space that has been adapted to the particular linear symbol present in the equation. The contraction is proved by using microlocal analysis and a trilinear estimate that is shown via the \([k; Z]\)-multiplier norm method developed by Terence Tao.

Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces

We give resolvent estimates near zero energy on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations, using a uniform microlocal Fredholm analysis framework.

Compressed sensing two-dimensional Bragg scatter imaging.

Here we introduce a new reconstruction technique for two-dimensional Bragg scattering tomography (BST), based on the Radon transform models of Webber and Miller [Inverse Probl. Imaging15, 683 (2021).10.3934/ipi.2021010]. Our method uses a combination of ideas from multibang control and microlocal analysis to construct an objective function which can regularize the BST artifacts; specifically the boundary artifacts due to sharp cutoff in sinogram space (as observed in [arXiv preprint, arXiv:2007.00208 (2020)]), and artifacts arising from approximations made in constructing the model used for inversion. We then test our algorithm in a variety of Monte Carlo (MC) simulated examples of practical interest in airport baggage screening and threat detection. The data used in our studies is generated with a novel Monte-Carlo code presented here. The model, which is available from the authors upon request, captures both the Bragg scatter effects described by BST as well as beam attenuation and Compton scatter.

A new microlocal analysis of hyperfunctions

In this work we study microlocal regularity of hyperfunctions defining in this context a class of generalized FBI transforms first introduced for distributions by Berhanu and Hounie. Using a microlocal decomposition of a hyperfunction and the generalized FBI transforms we were able to characterize the wave-front set of hyperfunctions according to several types of regularity. The microlocal decomposition allowed us to recover and generalize both classical and recent results and, in particular, we proved for differential operators with real-analytic coefficients that if the elliptic regularity theorem regarding any reasonable regularity holds for distributions, then it is automatically true for hyperfunctions.

Microlocal Analysis and Characterization of Sobolev Wavefront Sets Using Shearlets

Sobolev wavefront sets and 2-microlocal spaces play a key role in describing and analyzing the singularities of distributions in microlocal analysis and solutions of partial differential equations. Employing the continuous shearlet transform to Sobolev spaces, in this paper we characterize the microlocal Sobolev wavefront sets, the 2-microlocal spaces, and local Hölder spaces of distributions/functions. We then establish the connections among Sobolev wavefront sets, 2-microlocal spaces, and local Hölder spaces through the continuous shearlet transform.

Analytic Berezin–Toeplitz operators

We introduce new tools for analytic microlocal analysis on Kähler manifolds. As an application, we prove that the space of Berezin–Toeplitz operators with analytic contravariant symbol is an algebra. We also give a short proof of the Bergman kernel asymptotics up to an exponentially small error.

Classical and microlocal analysis of the x-ray transform on Anosov manifolds

We complete the microlocal study of the geodesic X-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou and pursued by Guillarmou and the second author. We prove new stability estimates and clarify some properties of the operator $\Pi_m$, the generalized X-ray transform. These estimates rely on a refined version of the Livsic theorem for Anosov flows, especially on a new quantitative finite time Livsic theorem.

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

In this note we consider the nonlinear heat equation associated to the fractional Hermite operator H^beta =(-Delta +|x|^2)^beta , 0<beta le 1. We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the Hörmander class S^m_{0,0}, min mathbb {R}.

Microlocal Analysis of Generalized Radon Transforms from Scattering Tomography

Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of $L^2$ functions of compact support over surfaces of revolution of $C^{\infty}$ curves $q$. We show that the Radon transforms are elliptic Fourier integral operators (FIO) and provide an analysis of the left projections $\Pi_L$. Our main theorem shows that $\Pi_L$ satisfies the semiglobal Bolker assumption if and only if $g=q'/q$ is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the generalized Radon FIO. Our theory has specific applications of interest in Emission Compton Scattering Tomography (ECST) and Bragg Scattering Tomography (BST). We show that the ECST and BST integration curves satisfy the semiglobal Bolker assumption and provide simulated reconstructions from ECST and BST data. Additionally, we give example “sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artifacts. The observed artifacts in reconstruction are shown to align exactly with our predictions.

Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation

<p style='text-indent:20px;'>We are concerned with the transmission problem of nonlinear viscoelastic waves in a heterogeneous medium, establishing the well-posedness of solutions and the exponential stability of the related energy functional. We introduce an auxiliary problem to prove the exponential stability and the proof combines an observability inequality and microlocal analysis tools.