Pseudodifferential Analysis Research Articles

Large-amplitude modulation of periodic traveling waves

<p style='text-indent:20px;'>We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudodifferential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.</p>

Pseudo-differential analysis of bounded linear operators from $$L^{p_1}({\mathbb S}^1)$$ into $$L^{p_2}({\mathbb S}^1)$$

In this paper, we first show that every bounded linear operator T from $$ L^{p_{1}}({\mathbb S}^1) $$ into $$ L^{p_{2}}({\mathbb S}^1) $$ for $$1< p_{1}, p_{2}<\infty $$ is a pseudo-differential operator and give a formula for its symbol $$ \sigma $$ . Using the pseudo-differential representation of T, we offer necessary and sufficient conditions on the symbols $$ \sigma $$ to find the symbol of the adjoint $$ T^{*}$$ . We present necessary and sufficient conditions on the symbols $$ \sigma $$ to ensure that their corresponding bounded linear operators from $$ L^{2}({\mathbb S}^1) $$ into $$ L^{2}({\mathbb S}^1)$$ are self-adjoint, compact or compact self-adjoint. As applications, in case $$ \sigma $$ is a real-valued function, we show that the corresponding bounded linear operator T is self-adjoint if and only if $$ \sigma $$ depends only on one variable. Also, we prove that every compact operator from $$ L^{2}({\mathbb S}^1) $$ into $$ L^{2}({\mathbb S}^1) $$ is the product of two compact operators and give some formulas for the symbols of compact and compact self-adjoint operators.

Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian

We consider the Schrödinger operator with constant magnetic field defined on the half-plane with a Dirichlet boundary condition, H0, and a decaying electric perturbation V. We study the Spectral Shift Function (SSF) associated to the pair (H0+V,H0) near the Landau levels, which are thresholds in the spectrum of H0. For perturbations of a fixed sign, we estimate the SSF in terms of the eigenvalue counting function for certain compact operators. If the decay of V is power-like, then using pseudodifferential analysis, we deduce that there are singularities at the thresholds and we obtain the corresponding asymptotic behavior of the SSF. Our technique gives also results for the Neumann boundary condition.

Nonharmonic Analysis of Boundary Value Problems Without WZ Condition

In this work we continue our research on nonharmonic analysis of boundary value problems as initiated in [8]. There, we assumed that the eigenfunctions of the model operator on which the construction is based do not have zeros. In this paper we have weakened this condition extending the applicability of the developed pseudo-differential analysis. Also, we do not assume that the underlying set Omega is bounded.

Complex Absorbing Potential Method for the Perturbed Dirac Operator

The Complex Absorbing Potential (CAP) method is widely used to compute resonances in quantum chemistry, both for nonrelativistic and relativistic Hamiltonians. In the semiclassical limit ℏ → 0 we consider resonances near the real axis and we establish the CAP method rigorously for the perturbed Dirac operator by proving that individual resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs are based on pseudodifferential operator theory and microlocal analysis.

Aspects of the zeta function originating from pseudodifferential analysis

Pseudodifferential analysis helps linking the one-dimensional and two-dimensional analyses in more than one way. Taking benefit from special features of the two-dimensional case, in particular the fact that homogeneous distributions, invariant under the action by linear changes of coordinates of \(SL(2,{\mathbb Z})\), are essentially disguised versions of nonholomorphic modular forms, we are led to introducing interesting two-dimensional distributions and some associated one-dimensional parts. This results in a collection of necessary and sufficient conditions for the Riemann hypothesis to hold, some of which, but not all, are of a more or less classical type.

Mathematical analysis as required by a tachyonic observer

We rely on an analysis of initial states of the wave function associated with the simplest relativistic particle—the Weyl neutrino—to construct two series of representations of \documentclass[12pt]{minimal}\begin{document}$SL(2,\mathbb {R})$\end{document}SL(2,R), or of the twofold cover of this group, by operators acting on scalar functions defined on the real line. The first one, which depends on a parameter p = 0, 1, …, contains the usual one-dimensional metaplectic, or oscillator, representation. The second one, a series of representations no longer unitary but, for certain values of the parameter, pseudo-unitary with respect to some non-degenerate indefinite pseudo-scalar product, is built in a comparable way, only exchanging the time coordinate with one of the spatial ones. The first series of representations was originally introduced in connection with automorphic pseudodifferential analysis; the second one is new, except for one value of the (continuous, in this case) parameter ρ, in which case it coincides with the recently introduced anaplectic representation. For each value of the parameter λ (= p or ρ), the basic operators Q and P from the usual Heisenberg pair give way to a new pair (Q, Pλ) and to another analysis of functions on the real line, with as rich a collection of (most of the time totally new) symmetries as the usual one. The unusual set of coordinates in Minkowski's space under consideration can be regarded as corresponding to a “tachyonic observer,” a notion which does not require venturing into debatable questions of Physics. On the other hand, the paper indicates in a special example the way one may recover well-posedness (in a non-classical sense) for an initial-value problem with data on a timelike surface.

Coherent states, pseudodifferential analysis and arithmetic

Basic questions regarding families of coherent states include describing some constructions of such and the way they can be applied to operator theory or partial differential equations. In both questions, pseudodifferential analysis is important. Recent developments indicate that they can contribute to methods in arithmetic, especially modular form theory.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

A Stochastic Differential Equation Model with Jumps for Fractional Advection and Dispersion

The path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation (SDE) driven by Brownian noise. This model may not be adequate for highly heterogeneous media. This paper extends the model to a general SDE driven by a Levy noise. Particle paths follow a Markov process with long jumps. Their transition probability density solves a forward equation derived here via pseudo-differential operator theory and Fourier analysis. In particular, the SDE with stable driving noise has a space-fractional advection-dispersion equation (fADE) with variable coefficients as the forward equation. This result provides a stochastic solution to anomalous diffusion models, and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations.

Composition formulas in the Weyl calculus

In pseudodifferential analysis, the usual composition formula, which has asymptotic value, extends that valid for differential operators. The one developed here is based instead on the decomposition of symbols (functions in R n × R n ) as integral superpositions of homogeneous ones, of degrees lying on the complex line with real part − n. It extends the one known in the one-dimensional case in connection with automorphic pseudodifferential analysis.

Projective pseudodifferential analysis and harmonic analysis

We consider pseudodifferential operators on functions on R n + 1 which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. The symbols of such restrictions can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space G n / H n = SL ( n + 1 , R ) / GL ( n , R ) , and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space P n ( R ) : these spaces are the representation spaces of the maximal degenerate series ( π i λ , ε ) of G n . This new approach to the quantization of G n / H n , already considered by other authors, has several advantages: as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of L 2 ( G n / H n ) under the quasiregular action of G n . We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation π i λ , ε .

Edge problems on configurations with model cones of different dimensions

Elliptic equations on configurations W = W-1 boolean OR (. . .) boolean OR W-N with edge Y and components W-j of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here edges. Starting from edge-degenerate operators on Wj, j = 1, . . . , N, we construct an algebra with extra 'transmission' conditions on Y that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator- valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on WY. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics

EDGE OPERATORS WITH CONDITIONS OF TOEPLITZ TYPE

Ellipticity of operators on a manifold with edges can be treated in the framework of a calculus of -block matrix operators with trace and potential operators on the edges. The picture is similar to the pseudodifferential analysis of boundary-value problems. The extra conditions satisfy an analogue of the Shapiro–Lopatinskij condition, provided a topological obstruction for the elliptic edge-degenerate operator in the upper left corner vanishes; this is an analogue of a condition of Atiyah and Bott in boundary-value problems. In general, however, we need global projection data, similarly to global boundary conditions, known for Dirac operators or other geometric operators. The present paper develops a new calculus with global projection data for operators on manifolds with edges. In particular, we show the Fredholm property in a suitable scale of spaces and construct parametrices within the calculus.

Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part II:L2(Ω)-estimates

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in , with C2-boundary subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H1(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H1-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C2-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H2-solutions, and ultimately for H1-solutions. The passage from H2- to H1-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H1-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H1-solutions satisfying over-determined boundary conditions, and—above all—(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions on the triple {Ω,Γ0,Γ1} that arise in the corresponding problems for second-order hyperbolic equations. The most general result, with weakest geometrical conditions, is, in fact, deferred to Section 9. Sections 1 through 8 provide the main body of our treatment with one vector field under a preliminary working geometrical condition, which is then removed in Section 9, by use of two suitable vector fields. The second and final goal of this paper is to shift the Carleman estimates (Hence, the continuous observability/stabilization inequalities) by one unit downward to the lower L2(Ω)-level. This is accomplished in Section 10 by means of pseudo-differential analysis, and accordingly, it contains lower-order terms. Applications of these L2(Ω)-Carleman estimate includes a new uniform stabilization of the conservative Schrödinger equation in the state space L2(Ω), by an attractive boundary feedback.

Pseudodifferential analysis on conformally compact spaces

Part 1. Fredholm theory for $0$-pseudodifferential operators: Review of basic objects of $0$-geometry The small $0$-calculus and the $0$-calculus with bounds The $b$-$c$-calculus on an interval The reduced normal operator Weighted $0$-Sobolev spaces Fredholm theory for $0$-pseudodifferential operators Part 2. Algebras of $0$-pseudodifferential operators of order $0$: $C^*$-algebras of $0$-pseudodifferential operators $\Psi^*$-algebras of $0$-pseudodifferential operators Appendix A. Spaces of conormal functions Bibliography Notations Index.

Thermophysical property determination of high temperature alloys by thermal analysis

Differential scanning calorimetric measurements to determine solidus and liquidus temperatures and latent heat of fusion of two high temperature materials, PWA1484 and an experimental gamma titanium aluminide alloy, are presented. The solidus and liquidus temperatures of PWA1484 are 1340 and 1404°C. The solidus and liquidus temperatures of the titanium aluminide alloy are 1453 and 1522°C. Solidus and liquidus temperatures determined from actual heating and cooling curves, which were measured using imbedded thermocouples and analyzed by a pseudo-differential thermal analysis technique are found to be in good agreement with the differential scanning calorimetric measurements.

Pseudodifferential analysis on continuous family groupoids

We study properties and representations of the convolution algebra and the algebra of pseudodifferential operators associated to a continuous family groupoid. We show that the study of representations of the algebras of pseudodifferential operators of order zero completely reduces to the study of the representations of the ideal of regularizing operators. This recovers the usual boundedness theorems for pseudodifferential operators of order zero. We prove a structure theorem for the norm completions of these algebras associated to groupoids with invariant filtrations. As a consequence, we obtain criteria for an operator to be compact or Fredholm. We end with a discussion of the significance of these results to the index theory of operators on certain singular spaces. For example, we give a new approach to the question of the existence of spectral sections for operators on coverings of manifolds with boundary. We expect that our results will also play a role in the analysis on more general singular spaces.

From pseudodifferential analysis to modular form theory

From pseudodifferential analysis to modular form theory

A. Unterberger and H. Upmeier Pseudodifferential analysis on symmetric cones (CRC Press, Boca Raton–New York–London–Tokyo, 1996), 216pp., 0 8493 7873 7, $59.95.

A. Unterberger and H. Upmeier Pseudodifferential analysis on symmetric cones (CRC Press, Boca Raton–New York–London–Tokyo, 1996), 216pp., 0 8493 7873 7, $59.95.

Sharp trace estimates of solutions to Kirchhoff and Euler-Bernoulli equations

This paper gives sharp estimates of second-order traces of solutions to mixed problems for Kirchhoff equations and Euler-Bernoulli equations with second- and third-order nonhomogeneous boundary conditions. The approach uses pseudodifferential analysis. These sharp trace regularity results have, among others, important implications in corresponding exact controllability/uniform stabilization theories; they allow the removal of geometrical conditions made in prior literature.