Principal Symbol Research Articles

The Cauchy problem for operators with injective symbol in the Lebesgue space L 2 in a domain

Let D be a bounded domain in ℝn (n ≥ 2) with infinitely smooth boundary ∂D. We give some necessary and sufficient conditions for the Cauchy problem to be solvable in the Lebesgue space L2(D) in D for an arbitrary differential operator A having an injective principal symbol. Furthermore, using bases with double orthogonality, we construct Carleman’s formula that restores a (vector-)function in L2(D) from the Cauchy data given on a relatively open connected set Γ ⊂ ∂D and the values Au in D whenever the data belong to L2(Γ) and L2(D) respectively.

Sur l’existence d’une solution ramifiée pour des équations de Fuchs à caractéristique simple

The aim of this paper is to construct a holomorphic solution, ramified around a simple characteristic hypersurface, for some linear Fuchsian equation of order $m\geq 1$. We consider an operator $L$, holomorphic in a neighborhood of the origin in ${\mathbb {C}}_t\times {\mathbb {C}}_x^n$, of the form $L=tA+B$ where $A$ and $B$ are linear partial differential operators of order $m$ and $m-1$, and where $A$ has a simple characteristic hypersurface transverse to $S:t=0$. Under an assumption linking the principal symbols of $A$ and $B$, the question is reduced to the study of an integro-differential Fuchsian equation with an additional variable $z$ that describes the universal covering of a pointed disk. It is an equation where terms like $t^lD_t^hD_x^\alpha (tD_t+1)^{-1}D_z^{-q}, l,h,q\in \mathbb {N}, \alpha \in \mathbb {N}^n$ with $l\leq 1$ and $h+|\alpha |\leq l+q$ appear. The problem is solved by the fixed-point theorem with appropriate estimations in a Banach space.

The Algebraic Integrability of the Quantum Toda Lattice and the Radon Transform

We study the maximal commutative ring of partial differential operators which includes the quantum completely integrable system defined by the quantum Toda lattice. Kostant shows that the image of the generalized Harish-Chandra homomorphism of the center of the enveloping algebra is commutative (Kostant in Invent. Math. 48:101–184, 1978). We demonstrate the commutativity of the ring of partial differential operators whose principal symbols are \({\bar{N}}\) -invariant. Our commutative ring includes the commutative system of Kostant (Invent. Math. 48:101–184, 1978). The main tools in this paper are Fourier integral operators and Radon transforms.

Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy $${\sigma=(\sigma_\psi,\sigma_\wedge)}$$ , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol $${\sigma_\wedge}$$ which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems.

Remarks on the Levi conditions for differential systems

In this paper we prove two results on the Levi conditions for weakly hyper- bolic systems with characteristics of constant multiplicities. A first result concerns scalar operators: we prove that Levi conditions defined by the second author in (29) are equivalent to the usual Levi conditions for scalar operator. A second result concerns systems whose principal symbol has a Jordan form made of a large number of 2 £ 2 blocks. For these systems we express the first Levi condition via an invariant constructed from the sub-characteristic matrix. Moreover we show that this condition is necessary for the C 1 well-posedness.

The Principal Symbols in A Vision

The Principal Symbols in A Vision

Motion compensated local tomography

In this paper we develop local tomography (LT) for image reconstruction from motion contaminated data. It is assumed that motion is known. We propose a new LT function fΛ, which is related to an original object f via an operator : . Because of motion, may fail to be a pseudo-differential operator (PDO). We obtain the conditions that guarantee that is a PDO. Under these conditions, similarly to the classical LT in is a PDO of order 1. Computation of fΛ depends on a weight function Φ. We show that Φ can be chosen in such a way that the operator has principal symbol |ξ|. This result has an interesting corollary for conventional exact reconstruction. It suggests a novel frequency-split approach to finding f from motion contaminated data. In practice tomographic data are discrete, and derivatives are usually replaced by their mollified analogs. We consider how mollification affects the singularities of the LT function fΛ. Using this approach we develop an algorithm for finding values of jumps of f using LT. We also consider various aspects of numerical implementation of LT and show the results of numerical experiments.

Inverse Scattering Problem for a Two Dimensional Random Potential

We study an inverse problem for the two-dimensional random Schrodinger equation (Δ + q + k2)u = 0. The potential q(x) is assumed to be a Gaussian random function whose covariance operator is a classical pseudodifferential operator. We show that the backscattered field, obtained from a single realization of the random potential q, determines uniquely the principal symbol of the covariance operator of q. The analysis is carried out by combining harmonic and microlocal analysis with stochastic methods.

The double exponential map on spaces of constant curvature

In the article, we derive an explicit formula for the double exponential map on spaces of constant curvature. In addition, we consider some applications of the resulting formula to computing the principal symbol of the product of two pseudodifferential operators on a manifold with connection.

Spreading of Lagrangian Regularity on Rational Invariant Tori

Let P h be a self-adjoint semiclassical pseudodifferential operator on a manifold M such that the bicharacteristic flow of the principal symbol on T*M is completely integrable and the subprincipal symbol of P h vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes u h of P h . We show that on a nondegenerate rational invariant torus, Lagrangian regularity of u h (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional “diffractive” manner. In particular, in addition to propagating along null bicharacteristics, regularity fills in the interiors of small annular tubes of bicharacteristics.

On the Cauchy Problem for Hyperbolic Operators with Nearly Constant Coefficient Principal Part

In this paper we shall deal with operators whose principal symbols can be microlocally transformed to symbols depending only on the fiber variables by homogeneous canonical transformations. We call such operators hyperbolic operators with nearly constant coefficient principal part. Operators with constant coefficient principal part and operators with involutive characteristics belong to this class of operators. We shall give a necessary and sufficient for the Cauchy problem to be C∞ well-posed under some additional assumptions. Namely, we shall generalize condition and prove that the generalized Levi is necessary and sufficient for the Cauchy problem to be C∞ well-posed.

On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness

In this paper, we prove that for non-effectively hyperbolic operators with smooth double characteristics exhibiting a Jordan block of size 4 on the double manifold, the Cauchy problem is well-posed in the Gevrey 5 class, beyond the generic Gevrey class 2 (see, e.g., [5]). Moreover, we show that this value is optimal, due to certain geometric constraints on the Hamiltonian flow of the principal symbol. These results, together with results already proved, give a complete picture of the well-posedness of the Cauchy problem around hyperbolic double characteristics.

Analytic Hypoellipticity in the Presence of Lower Order Terms

We consider a second order operator with analytic coefficients whose principal symbol vanishes exactly to order two on a symplectic real analytic manifold. We assume that the first (non degenerate) eigenvalue vanishes on a symplectic submanifold of the characteristic manifold. In the C ∞ framework this situation would mean a loss of 3/2 derivatives (see Helffer, 1977). We prove that this operator is analytic hypoelliptic. The main tool is the FBI transform. A case in which C ∞ hypoellipticity fails is also discussed.

A general effective Hamiltonian method

We perform a general reduction scheme that can be applied in particular to the spectral study of operators of the type P=P(x,y,hD_x,D_y) as h tends to zero. This scheme permits to reduce the study of P to the one of a semiclassical matrix operator of the type A=A(x,hD_x) . Here, for any fixed (x,\xi )\in\R^n , the eigenvalues of the principal symbol a(x,\xi ) of A are eigenvalues of the operator P(x,y,\xi ,D_y) .

$${\mathcal{R}}$$ -boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class (\({\mathcal{HT}}\)) and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with \({\mathcal{R}}\) -bounded symbols, yielding by an iteration argument the \({\mathcal{R}}\) -boundedness of λ(A−λ)−1 in \(\Re(\lambda) \geq \gamma\) for some \(\gamma \in {\mathbb{R}}\) . To this end, elements of a symbolic and operator calculus of pseudodifferential operators with \({\mathcal{R}}\) -bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on \({\mathbb{R}}^{d}\) with operator valued coefficients.

Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

We study the microlocal kernel of h-pseudodifferential operators Op h ( p ) − z , where z belongs to some neighborhood of size O ( h ) of a critical value of its principal symbol p 0 ( x , ξ ) . We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow H p 0 . First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C ∞ category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with given data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation.

ANISOTROPIC STABLE LEVY COPULA PROCESSES — ANALYTICAL AND NUMERICAL ASPECTS

We consider the valuation of derivative contracts on baskets of risky assets whose prices are Lévy-like Feller processes of tempered stable type. The dependence among the marginals' jump structure is parametrized by a Lévy copula. For marginals of regular, exponential Lévy type in the sense of Ref. 7 we show that the infinitesimal generator [Formula: see text] of the resulting Lévy copula process is a pseudo-differential operator whose principal symbol is a distribution of anisotropic homogeneity. We analyze the jump measure of the corresponding Lévy copula processes. We prove that the domains of their infinitesimal generators [Formula: see text] are certain anisotropic Sobolev spaces. In these spaces and for a large class of Lévy copula processes, we prove a Gårding inequality for [Formula: see text]. We design a wavelet-based dimension-independent tensor product discretization for the efficient numerical solution of the parabolic Kolmogorov equation [Formula: see text] arising in valuation of derivative contracts under possibly stopped Lévy copula processes. In the wavelet basis, diagonal preconditioning yields a bounded condition number of the resulting matrices.

Stability for a class of nonlinear pseudo-differential equations

We study a class of nonlinear evolutionary equations generated by a pseudo-differential operator with the elliptic principal symbol and with nonlinearities of the form G ( u x ) where c η 2 ≤ G ( η ) ≤ C η 2 for large | η | . We demonstrate existence of a universal absorbing set, and a compact attractor, and show that the attractor is of a finite Hausdorff dimension. The stabilization mechanism is similar to the nonlinear saturation well known for the Kuramoto–Sivashinsky equation.

Sharp semiclassical estimates for the number of eigenvalues below a totally degenerate critical level

We consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than E for elliptic operators in L 2 ( R d ) . We describe a method of obtaining remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the intervals [ E ′ ; E ′ + h ] , where E ′ is close to E. If the volume of this region is O ( h ) , then we obtain remainder estimates O ( h 1 − d ) with no assumptions on the Hessian of the principal symbol at the critical level. Moreover we do not assume that the coefficients are smooth—all results hold if second order derivatives of coefficients are Hölder continuous.

Completely Integrable Systems Associated with Classical Root Systems

We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the Bn-invariants. We review conditions supporting the conjecture and give a new condition assuring it.