Pseudodifferential Operators Of Order Research Articles

The scattering matrix for 0th order pseudodifferential operators

The scattering matrix for 0th order pseudodifferential operators

Connes' integration and Weyl's laws

This paper deal with some questions regarding the notion of integral in the framework of Connes's noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman-Solomyak's perturbation theory. We also give a soft proof of Birman-Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassical Weyl's law for Schroedinger operators. This is an easy consequence of the Birman-Schwinger principle. We thus get a neat link between noncommutative geometry and semiclassical analysis.

The discrete spectrum of the Neumann-Poincaré operator in 3D elasticity

For the Neumann-Poincaré (double layer potential) operator in the three-dimensional elasticity we establish asymptotic formulas for eigenvalues converging to the points of the essential spectrum and discuss geometric and mechanical meaning of coefficients in these formulas. In particular, we establish that for any body, there are infinitely many eigenvalues converging from above to each point of the essential spectrum. On the other hand, if there is a point where the boundary is concave (in particular, if the body contains cavities) then for each point of the essential spectrum there exists a sequence of eigenvalues converging to this point from below. The reasoning is based upon the representation of the Neumann-Poincaré operator as a zero order pseudodifferential operator on the boundary and the earlier results by the author on the eigenvalue asymptotics for polynomially compact pseudodifferential operators.

Fractional‐order operators on nonsmooth domains

AbstractThe fractional Laplacian , , and its generalizations to variable‐coefficient ‐order pseudodifferential operators , are studied in ‐Sobolev spaces of Bessel‐potential type . For a bounded open set , consider the homogeneous Dirichlet problem: in , in . We find the regularity of solutions and determine the exact Dirichlet domain (the space of solutions with ) in cases where has limited smoothness , for , . Earlier, the regularity and Dirichlet domains were determined for smooth by the second author, and the regularity was found in low‐order Hölder spaces for by Ros‐Oton and Serra. The ‐results obtained now when are new, even for . In detail, the spaces are identified as ‐transmission spaces , exhibiting estimates in terms of near the boundary.The result has required a new development of methods to handle nonsmooth coordinate changes for pseudodifferential operators, which have not been available before; this constitutes another main contribution of the paper.

Discrete spectrum of zero order pseudodifferential operators

Discrete spectrum of zero order pseudodifferential operators

Global well–posedness of a class of singular hyperbolic Cauchy problems

The goal of this paper is to establish a global well–posedness, cone condition and loss of regularity for singular hyperbolic equations with coefficients in $$L^1((0,T];C^\infty (\mathbb {R}^n)) \cap C^1((0,T];C^\infty (\mathbb {R}^n))$$ and Cauchy data in an appropriate Sobolev space tailored to a metric on the phase space. The coefficients are unbounded near the singular hyperplane $$t=0$$ and polynomially growing as $$|x| \rightarrow \infty .$$ The singular behavior is characterized by the blow–up rate of the coefficients and their first t-derivatives near $$t=0.$$ In order to study the interplay of the singularity in t and unboundedness in x, we consider a class of metrics on the phase space. Our methodology relies on the use of the Planck function associated to the metric to subdivide the extended phase and to define an infinite order pseudodifferential operator for the conjugation. We also give some counterexamples.

Dynamics of Resonances for 0th Order Pseudodifferential Operators

We study the dynamics of resonances of analytic perturbations of 0th order pseudodifferential operators P(s). In particular, we prove a Fermi golden rule for resonances of P(s) at embedded eigenvalues of $$P=P(0)$$ . We answer the question on the generic absence of eigenvalues asked by Colin de Verdière (Anal PDE 13:1521–1537, 2020). We also study the dynamics of eigenvalues of $$P+it\Delta $$ as the eigenvalues converge to simple eigenvalues of P. The 0th order pseudodifferential operators we consider satisfy natural dynamical assumptions and are used as microlocal models of internal waves.

Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators

<p style='text-indent:20px;'>Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations.</p>

Global $${\mathcal {M}}-$$Hypoellipticity, Global $${\mathcal {M}}-$$Solvability and Perturbations by Lower Order Ultradifferential Pseudodifferential Operators

We introduce a new class of ultradifferentiable pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity, in ultradifferentiable classes, with a finite loss of derivatives of a system of pseudodifferential operators, is stable under perturbations by lower order pseudodifferential operators whose order depends on the loss of derivatives. The key point in our study is our definition of loss of derivatives. We also give an easy proof of the fact that if a system of pseudodifferential operators is globally $${\mathcal {M}}$$ -hypoelliptic then its transpose is globally solvable in $$D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right) $$ . Finally we present an application of our results.

Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results

In this paper we establish uniqueness theorems for the noncommutative residue and the canonical trace on pseudodifferential operators on noncommutative tori of arbitrary dimension. The former is the unique trace up to constant multiple on integer order pseudodifferential operators.The latter is the unique trace up to constant multiple on non-integer order pseudodifferential operators. This improves previous uniqueness results by Fathizadeh-Khalkhali, Fathizadeh-Wong, and L\'evy-Neira-Paycha.

Microlocal analysis of forced waves

We use radial estimates for pseudodifferential operators to describe long time evolution of solutions to $ i u_t - P u = f $ where $ P $ is a self-adjoint 0th order pseudodifferential operator satisfying hyperbolic dynamical assumptions and where $ f $ is smooth. This is motivated by recent results of Colin de Verdi\`ere and Saint-Raymond [arXiv:1801.05582] concerning a microlocal model of internal waves in stratified fluids.

Normal Form Coordinates for the KdV Equation Having Expansions in Terms of Pseudodifferential Operators

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudodifferential operator of order 0 with principal part given by the Fourier transform and (2) the pullback of the KdV Hamiltonian is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the KdV equation under small, quasi-linear perturbations.

Global hypoellipticity for first-order operators on closed smooth manifolds

The main goal of this paper is to address global hypoellipticity issues for the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \in \mathbb{T} \times M$, $\mathbb{T}$ is the one-dimensional torus, $M$ is a closed manifold and $C(t,x,D_x)$ is a first order pseudo-differential operator on $M$, smoothly depending on the periodic variable $t$. In the case of separation of variables, namely, $C(t,x,D_x) = a(t)p(x,D_x)+ib(t)q(x,D_x)$, we give necessary and sufficient conditions for the global hypoellipticity of $L$. In particular, we show that, under suitable conditions, the famous (P) condition of Niremberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of $L$.

Spectral asymptotics for infinite order pseudo-differential operators

We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.

Microlocal properties of scattering matrices

ABSTRACTWe consider the scattering theory for a pair of operators H0 and H = H0 + V on L2(M, m), where M is a Riemannian manifold, H0 is a multiplication operator on M, and V is a pseudodifferential operator of order − μ, μ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.

The broken ray transform in $n$ dimensions with flat reflecting boundary

We study the broken ray transform on $n$-dimensional Euclidean domains where the reflecting parts of the boundary are flat and establish injectivity and stability under certain conditions. Given a subset $E$ of the boundary $\partial \Omega$ such that $\partial \Omega \setminus E$ is itself flat (contained in a union of hyperplanes), we measure the attenuation of all broken rays starting and ending at $E$ with the standard optical reflection rule applied to $\partial \Omega \setminus E$. By localizing the measurement operator around broken rays which reflect off a fixed sequence of flat hyperplanes, we can apply the analytic microlocal approach of Frigyik, Stefanov, and Uhlmann for the ordinary ray transform by means of a local path unfolding. This generalizes the author's previous result for the square, although we can no longer treat reflections from corner points. Similar to the result for the two dimensional square, we show that the normal operator is a classical pseudo differential operator of order -1 plus a smoothing term with $C_{0}^{\infty}$ Schwartz kernel.

Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions

The adiabatic groupoid Gad of a smooth groupoid G is a deformation relating G with its algebroid. In a previous work, we constructed a natural action of R on the C*-algebra of zero order pseudodifferential operators on G and identified the crossed product with a natural ideal J(G) of C⁎(Gad). In the present paper we show that C⁎(Gad) itself is a pseudodifferential extension of this crossed product in a sense introduced by Saad Baaj. Let us point out that we prove our results in a slightly more general situation: the smooth groupoid G is assumed to act on a C*-algebra A. We construct in this generalized setting the extension of order 0 pseudodifferential operators Ψ(A,G) of the associated crossed product A⋊G. We show that R acts naturally on Ψ(A,G) and identify the crossed product of A by the action of the adiabatic groupoid Gad with an extension of the crossed product Ψ(A,G)⋊R. Note that our construction of Ψ(A,G) unifies the ones of Connes (case A=C) and of Baaj (G is a Lie group).

Cauchy problems related to integrable deformations of pseudo differential operators

In this paper we discuss the solvability of two Cauchy problems in the pseudo differential operators. The first is associated with a set of pseudo differential operators of negative order, the prominent example being the set of strict integral operator parts of the different powers of a solution of the KP hierarchy. We show that it can be solved, provided the setting possesses a compatibility completeness. In such a setting all solutions of the KP hierarchy are obtained by dressing with the solution of the related Cauchy problem. The second Cauchy problem is slightly more general and links up with a set of pseudo differential operators of order zero or less. The key example here is the collection of integral operator parts of the different powers of a solution of the strict KP hierarchy. This system is solvable as soon as exponential and compatibility completeness holds. Also under these circumstances, all solutions of the strict KP hierarchy are obtained by dressing with the solution of the corresponding Cauchy problem.

The Broken Ray Transform on the Square

We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for \(C_{0}^{2}\) perturbations of a vanishing absorption parameter \(\sigma \equiv 0\). Given an open subset \(E\) of the boundary, we measure the attenuation of all broken rays starting and ending at \(E\) with the standard optical reflection rule applied to \(\partial \Omega {\setminus } E\). Using the analytic microlocal approach of Frigyik et al. for the X-ray transform on generic families of curves, we show injectivity via a path unfolding argument under suitable conditions on the available broken rays. Then we show that with a suitable decomposition of the measurement operator via smooth cutoff functions, the associated normal operator is a classical pseudo differential operator of order \(-1\), which leads to the desired result.

Localization results for zero order pseudodifferential operators

We show phase space localization at suitable energies for zero order pseudodifferential operators, implying non-propagation properties for the associated evolution groups.