Singular Pseudodifferential Operators Research Articles

Kipriyanov Singular Pseudodifferential Operators Generated by Bessel 𝕁-Transform

Kipriyanov Singular Pseudodifferential Operators Generated by Bessel 𝕁-Transform

The virial theorem for a class of singular pseudo-differential operators on $${\mathbb {R}}^n$$ R n

A quantum mechanical version of the virial theorem for singular pseudo-differential operators modelling relativistic Hamiltonians on \({\mathbb {R}}^n\) is established and nonexistence of eigenvalues of the singular pseudo-differential operators are derived using the virial theorem.

Semilinear geometric optics with boundary amplification

We study weakly stable semilinear hyperbolic boundary value problems with highly oscillatory data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency $\beta$ in the hyperbolic region. As a consequence of this degeneracy there is an amplification phenomenon: outgoing waves of amplitude $O(\eps^2)$ and wavelength $\eps$ give rise to reflected waves of amplitude $O(\eps)$, so the overall solution has amplitude $O(\eps)$. Moreover, the reflecting waves emanate from a radiating wave that propagates in the boundary along a characteristic of the Lopatinskii determinant. An approximate solution that displays the qualitative behavior just described is constructed by solving suitable profile equations that exhibit a loss of derivatives, so we solve the profile equations by a Nash-Moser iteration. The exact solution is constructed by solving an associated singular problem involving singular derivatives of the form $\partial_{x'}+\beta\frac{\partial_{\theta_0}}{\eps}$, $x'$ being the tangential variables with respect to the boundary. Tame estimates for the linearization of that problem are proved using a first-order calculus of singular pseudodifferential operators constructed in the companion article \cite{CGW2}. These estimates exhibit a loss of one singular derivative and force us to construct the exact solution by a separate Nash-Moser iteration. The same estimates are used in the error analysis, which shows that the exact and approximate solutions are close in $L^\infty$ on a fixed time interval independent of the (small) wavelength $\eps$. The approach using singular systems allows us to avoid constructing high order expansions and making small divisor assumptions.

A Priori Estimate for Solutions of Singular B–Elliptic Pseudodifferential Equations with Bessel ∂ B –Operators

Based on a special Fourier–Bessel transform introduced by Kipriyanov and Katrakhov, we consider singular pseudodifferential operators including linear singular differential operators. We introduce B-elliptic operators and construct regularizers for B-elliptic equations. A priori estimates are proved. Bibliography: 7 titles.

Full Fourier-Bessel transform and the algebra of singular pseudodifferential operators

The one-dimensional full Fourier-Bessel transform was introduced by I.A. Kipriyanov and V.V. Katrakhov on the basis of even and odd small (normalized) Bessel functions. We introduce a mixed full Fourier-Bessel transform and prove an inversion formula for it. Singular pseudodifferential operators are introduced on the basis of the mixed full Fourier-Bessel transform. This class of operators includes linear differential operators in which the singular Bessel operator and its (integer) powers or the derivative (only of the first order) of powers of the Bessel operator act in one of the directions. We suggest a method for constructing the asymptotic expansion of a product of such operators. We present the form of the adjoint singular pseudodifferential operator and show that the constructed algebra is, in a sense, a *-algebra.

Practical issues of reverse time migration: true amplitude gathers, noise removal and harmonic-source encoding

Recently, reverse-time migration (RTM) has drawn a lot of attention in the industry. Unlike one-way wave equation migration, RTM does not need to deal with the theory of singular pseudo-differential operators. A straightforward implementation of RTM correctly handles complex velocities and produces a complete set of acoustic waves (reflections, refractions, diffractions, multiples, evanescent waves, etc.). The RTM propagator also carries the correct propagation amplitude and imposes no dip limitations on the image. In the past, the strong migration artifacts and the intensive computational cost have been the two major problems that prevented RTM from being used in production. In this paper, we first formulate RTM based on inversion theory and then we address some solutions to suppress the low frequency migration artifacts. At the end, we propose harmonic-source migration as a way to improve the efficiency of delayed-shot RTM.

On a certain class of spherical harmonic functions and singular pseudo-differential operators

This article gives some results for the weighted spherical harmonic functions. Some relations are found between the singular integral operators generated by generalized shift operator and singular pseudo-differential operators. Finally, the boundedness of the pseudo-differential operators studied are shown.

Singular Pseudodifferential Operators, Symmetrizers, and Oscillatory Multidimensional Shocks

We introduce a calculus of singular pseudodifferential operators (SPOs) depending on wavelength ε and use them to solve three different types of singular quasilinear hyperbolic systems. Such systems arise in nonlinear geometric optics and also, for example, in the study of incompressible limits and of nonlinear wave equations with small nonlinear terms or small data. The SPOs act in both slow and fast variables and are singular not only because their symbols have finite regularity and depend on 1ε, but also because their derivatives fail to decay in the usual way in the dual variables. There is a necessarily crude calculus with large parameter (e.g., residual operators are just bounded on L2), but the calculus admits the proof of Garding inequalities and enables us to symmetrize and sometimes even diagonalize the singular systems being considered by microlocalizing simultaneously in both slow and fast variables. The paper culminates in a proof of the existence of oscillatory multidimensional shocks on a fixed time interval independent of the wavelength ε as ε→0. The use of SPOs allows us to eliminate the small divisor assumptions made in earlier work and also to construct more general oscillatory solutions in which elliptic boundary layers are present on one or both sides of the shock.

Some Remarks on singular pseudodifferential operators

Some Remarks on singular pseudodifferential operators

ON A CLASS OF ONE-DIMENSIONAL SINGULAR PSEUDODIFFERENTIAL OPERATORS

The paper is devoted to studying the method of the Sonin and Poisson transmutation operators for ordinary differential operators with variable coefficients. A new class of singular pseudodifferential operators is introduced which is closed under multiplication and which contains differential operators with singular coefficients, in particular, the Bessel operator. It is shown that after application of transmutation operators the original operator with variable coefficients (even a differential operator) is transformed into a classical pseudodifferential operator. The converse is also true: a pseudodifferential operator is transformed into a singular pseudodifferential operator. In all cases formulas are given for computing the symbols.Bibliography: 6 titles.