Analytic Pseudodifferential Operators Research Articles

Algebras of pseudo-differential operators acting on holomorphic Sobolev spaces

We search for pseudo-differential operators acting on holomorphic Sobolev spaces. The operators should mirror the standard Sobolev mapping property in the holomorphic analogues. The setting is a closed real-analytic Riemannian manifold, or Lie group with a bi-invariant metric, and the holomorphic Sobolev spaces are defined on a fixed Grauert tube about the core manifold. We find that every pseudo-differential operator in the commutant of the Laplacian is of this kind. Moreover, so are all the operators in the commutant of certain analytic pseudo-differential operators, but for more general tubes, provided that an old statement of Boutet de Monvel holds true generally. In the Lie group setting, we find even larger algebras, and characterize all their elliptic elements. These latter algebras are determined by global matrix-valued symbols.

A holomorphic mapping property of analytic pseudo-differential operators

We study the holomorphic extendibility of $\text{Op}(p)u$, when $p$ is an analytic symbol, and explicit information is available on the domains of holomorphic extendibility of both $p$ and $u$. By a contour deformation argument, we obtain a precise local estimate of the domain of holomorphy of $\text{Op}(p)u$ in terms of the information on $p$ and $u$.

Surprising examples of nonrational smooth spectral surfaces

In this paper we study necessary and sufficient algebro-geometric conditions for the existence of a nontrivial commutative subalgebra of rank in , a completion of the algebra of partial differential operators in two variables, which can be thought of as a simple algebraic analogue of the algebra of analytic pseudodifferential operators on a manifold. These are conditions on a projective (spectral) surface; they are encoded in a new notion of pre-spectral data. For smooth surfaces the sufficient conditions look especially simple. On a smooth projective surface there should exist 1) an ample integral curve with and ; 2) a divisor with , , , and . Amazingly, there are examples of such surfaces for which the corresponding commutative subalgebras do not admit isospectral deformations. Bibliography: 45 titles.

Foundation of symbol theory for analytic pseudodifferential operators, I

A new symbol theory for pseudodifferential operators in the complex analytic category is given. Here the pseudodifferential operators mean integral operators with real holomorphic microfunction kernels. The notion of real holomorphic microfunctions had been introduced by Sato, Kawai and Kashiwara by using sheaf cohomology theory. Symbol theory for those operators was partly developed by Kataoka and by the first author and it has been effectively used in the analysis of operators of infinite order. However, there was a missing part that links the symbol theory and the cohomological definition of operators, that is, the consistency of the Leibniz–Hörmander rule and the cohomological definition of composition for operators. This link has not been established completely in the existing symbol theory. This paper supplies the link and provides a cohomological foundation of the symbolic calculus of pseudodifferential operators.

Diophantine Tori and Weyl Laws for Non-selfadjoint Operators in Dimension Two

We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two, assuming that the classical flow of the unperturbed part is completely integrable. An asymptotic formula of Weyl type for the number of eigenvalues in a spectral band, bounded from above and from below by levels corresponding to Diophantine invariant Lagrangian tori, is established. The Weyl law is given in terms of the long time averages of the leading non-selfadjoint perturbation along the classical flow of the unperturbed part.

Asymptotic expansion for the widths of resonances in Born–Oppenheimer approximation

We develop a microlocal study for scalar classical analytic pseudo-differential operators of principal type. Next, we reduce a system of pseudo-differential operators to the normal form. Then we obtain asymptotic expansion and the exact exponential decay rate for the width of resonances arising from microlocal tunneling effect.

Analytic hypoellipticity and local solvability for a class of pseudo-differential operators with symplectic characteristics

p(x, ξ) ∼ pμ(x, ξ) + pμ−1(x, ξ) + . . . , where pμ−j(x, ξ) is positively homogeneous of degree μ− j with respect to ξ. We assume that the characteristic set Σ = p−1 μ (0) of P is a symplectic real analytic submanifold of T ∗(Ω)\0 of codimension 2d and that pμ vanishes exactly at the order m on Σ. As in Grusin [4], Sjostrand [11] and Metivier [8], we also assume that pμ−j vanishes at the order m− 2j on Σ for j ≤ m/2. C∞ and analytic hypoellipticity of this class of operators has been extensively studied by many mathematicians (see e.g., [1], [2], [4], [8], [9], [11], [13] and others). Among them Metivier [8] has proved analytic hypoellipticity of P by constructing a left parametrix when P is subelliptic with loss of m/2 derivatives. In this note, we study hypoellipticity and local solvability of P at a point where the above subellipticity condition is not satisfied. We shall then construct a system of analytic pseudo-differential operators on RN−d to which we can reduce the study of analytic hypoellipticity and local solvability of P . Typical examples of the operators are

Book Review: Analytic pseudodifferential operators for the Heisenberg group and local solvability

Book Review: Analytic pseudodifferential operators for the Heisenberg group and local solvability

Concatenations Applied to Analytic Hypoellipticity of Operators with Double Characteristics

We use the method of concatenations to get a sufficient condition for a class of analytic pseudodifferential operators with double characteristics to be analytic hypoelliptic. Introduction. The present paper is concerned with analytic hypoellipticity for operators on an N-dimensional real-analytic manifold i2, of the form (1) P(x, D) = Idp(x, D) + Q(x, D), where Id is the identity d x d matrix, p(x, () a scalar analytic symbol, homogeneous of degree m in (, and Q(x, D) a d x d matrix of classical analytic pseudodifferential operators of order m 1 (classical means that its total formal symbol is a series of homogeneous terms whose homogeneous degrees drop by integers). We assume that its principal symbol, p(x, (), is nonnegative everywhere, that it vanishes exactly of order two on its characteristic set L, and that L is a symplectic real-analytic submanifold of T*2 0. For such an operator, analytic hypoellipticity was already obtained in F. Treves [13] under the additional hypothesis that (2) P ( x, D) is hypoelliptic with loss of one derivative. Recently, several other studies have been made for the analytic hypoellipticity of similar operators. For example, in [7], G. Metivier extended the result of [13] to the operators with multiple characteristics assuming suitable hypoellipticity in case that the characteristic manifold is symplectic. Whereas, in [6], A. Grigis and L. P. Rothschild gave a necessary and sufficient condition for a class of operators with polynomial coefficients to be analytic hypoelliptic (see also [10] and [11] for a different approach to similar problems). On the other hand, in [3], L. Boutet de Monvel and F. Treves obtained a necessary and sufficient condition for P(x, D) to be hypoelliptic with loss of one derivative by means of the method of concatenations (introduced first by F. Treves in [12]). According to their work, we can associate with P(x, D) a sequence of operators p(v), P > 0, with the same principal as P(x, D), which satisfy certain relations connecting them, called concatenations. In terms of concatenations, the condition (2) can be restated by saying that the lower order part of every p(v) (i.e., the one Received by the editors March 25, 1983 and, in revised form, September 15, 1983. 1980 Mathematics Subject Classification. Primary 35B65.

Concatenations applied to analytic hypoellipticity of operators with double characteristics

We use the method of concatenations to get a sufficient condition for a class of analytic pseudodifferential operators with double characteristics to be analytic hypoelliptic.

Sharp estimates for analytic pseudodifferential operators and application to Cauchy problems

Sharp estimates for analytic pseudodifferential operators and application to Cauchy problems

Kernels and symbols of analytic pseudodifferential operators

Kernels and symbols of analytic pseudodifferential operators

Singular solutions for analytic pseudodifferential operators of principal type

Singular solutions for analytic pseudodifferential operators of principal type

Cauchy problem for analytic pseudo-differential operators

Cauchy problem for analytic pseudo-differential operators