Real Principal Type Research Articles

Inverse problems for real principal type operators

abstract: We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.

A Remark on the Essential Self-adjointness for Klein–Gordon-Type Operators

We discuss a new simplified proof of the essential self-adjointness for formally self-adjoint differential operators of real principal type with the null non-trapping condition, previously proved by Vasy (J Spectr Theory 10: 439–461, 2020) and Nakamura-Taira (Ann Henri Lebesgue 4: 1035–1059, 2021). For simplicity, here we discuss the second-order cases, i.e., Klein–Gordon-type operators only.

Propagation of polarization in transmission problems

For geometric systems of real principal type, we define a subprincipal symbol and derive a transport equation for polarizations which, in the scalar case, is a well-known equation of Duistermaat and H\"ormander. We apply the transport equation to propagation of polarization in transmission problems of elastodynamics, to interior bulk waves as well as to free (Rayleigh) surface waves. Using spectral factorizations of matrix polynomials having real spectrum, we establish reflection and refraction laws of polarizations at the boundary and at interior interfaces. The results are not limited to isotropic elasticity.

Remarks on the geodesically completeness and the smoothing effect on asymptotically Minkowski spacetimes

In this note, we study a geometric property of asymptotically Minkowski spacetimes and an analytic property of the wave operator. More precisely, our first main results show that asymptotically Minkowski spacetimes are geodesically complete under a null non-trapping condition. Secondly, we prove that Sobolev index of a real principal type estimate used in a previous work is optimal.

Essential self-adjointness of real principal type operators

We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.

A note on a class of Fuchsian operators

We study the propagation of singularities of a class of higher order pseudo-differential operators with a real principal symbol and with a characteristic variety which is the union of two smooth hypersurfaces having a non-involutive intersection. By a decoupling process we show that the operator, which is hitherto widely believed to be of non-involutive type, also has a propagation behavior of a real principal type operator. This decoupling also explains the branching of singularities phenomenon of these operators. Our methodology easily generalizes to systems.

Simultaneous triangularization of pseudo-differential systems

Using generalized Sylvester equation, we first develop a scheme for triangularizing a pseudo-differential system and prove a necessary and sufficient condition for factorizing the system into lower and upper triangular systems (LU factorization) independent of lower order terms. We then investigate simultaneous triangularization of a pair of matrix valued pseudo-differential operators under certain spectral conditions on their principal symbols. We also provide two extension results to real principal type systems using a Levi-like condition on the subprincipal symbol and singular symbol calculus.

Semiclassical second microlocal propagation of regularity and integrable systems

We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold X with respect to a Lagrangian submanifold of T*X. The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of T*ℝn, and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to Hormander’s theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g., eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.

Fundamental solutions for a class of non-elliptic homogeneous differential operators

We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be constructed in terms of radial averages and invariant distributions on the unit sphere.

Remarks on dispersive estimates and curvature

We investigate connections between certain dispersive estimates of a (pseudo) differential operator of real principal type and the number of non-vanishing curvatures of its characteristic manifold. More precisely, we obtain sharp thresholds for the range of Lebesgue exponents depending on the specific geometry.

Spectral Estimates for Degenerate Critical Levels

We establish spectral estimates at a critical energy level for h-pseudo-differential operators. Via a trace formula, we compute the contribution of isolated (nonextremum) critical points under a condition of "real principal type." The main result holds for all dimensions, for a singularity of any finite order and can be invariantly expressed in terms of the geometry of the singularity. When the singularities are not integrable on the energy surface the results are significative since the order w.r.t. h of the spectral distributions are bigger than in the regular setting.

Microlocal analytic smoothing effects for operators of real principal type

We are interested in the microlocal smoothing effect for operators of real principal type. On the initial value problem for a dispersive evolution equation, we study the fact that the sufficient decay of the initial data gives th e smoothness of the solution. We develop the theory of the FBI transform in order to transform our operator of real principal type into a simple operator of first order. S ince the smoothing effect is of global nature, our transformation is realized globall y along the bicharacteristics defined from the principal symbol of the operator.

Asymptotic approximation of degenerate fiber integrals

We study asymptotics of fiber integrals depending on a large parameter. When the critical fiber is singular, full-asymptotic expansions are established in two different cases: local extremum and isolated real principal type singularities. The main coefficients are computed and invariantly expressed. In the most singular cases, it is shown that the leading term of the expansion is related to invariant measures on the spherical blow-up of the singularity. The results can be applied to certain degenerate oscillatory integrals which occur in spectral analysis and quantum mechanics.

Lagrangian solutions to systems of real principal type

AbstractWe derive a transport equation for the principal symbol of a Fourier integral operator which solves a real principal type system. We apply this to Lamé's system of elastodynamics and obtain transport equations which remain valid at caustics. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a h-pseudo-differential operator on R n whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of “real principal type”. The new contribution to the trace formula is valid for all time in a compact subset of R but the result is modest since we have restrictions on the dimension.

Sharp L p Carleman estimates and unique continuation

We will present a unique continuation result for solutions of second order differential equations of real principal type P(x,D)u+V(x)u=0 with critical potential V in L n/2 (where n is the number of variables) across non-characteristic pseudo-convex hypersurfaces. To obtain unique continuation we prove L p Carleman estimates, this is achieved by constructing a parametrix for the operator conjugated by the Carleman exponential weight and investigating its L p -L p ' boundedness properties.

Boundary determination for an inverse problem in elastodynamics

We consider the unique determination of material properties of an inhomogeneous, isotropic, elastic object from measurements made at the surface. The behavior of the 3-dimensional object is modeled by solutions of the hear. hyperbolic system of equations for elastodynamics. The material properties of the object (its density and elastic properties) correspond to the (leading) coefficients of these diffrential equations. Modeling surface measurements by the Dirichlet-to-Neumann map on a finite time interval, we show here that the Dirichlet-to-Neumann map uniquely determines the density and elastic properties of the surface of the object. Solution of this problem leads to a proof in [R] of unique determination of certain properties(the wave speeds) of the interior of the object. Techniques used here include geometric optics. propagation of singularities for systems of real principal type, and Hamilton-Jacobi theory.

A symbolic construction of the forward fundamental solution of the wave operator

A calculus of polyhomogeneous paired Lagrangian distributions, associated to any two cleanly intersecting Lagrangain submanifolds, is constructed. The class is given an intrinsic characterisation using radial operators and a symbol calculus is developed. A class of pseudo—differential operators with singular symbols is developed within the calculus. This is used to give symbolic constructions of parametrices for operators of real principal type and paired Lagrangian distributions. The calculus is then applied to give a symbolic construction of the forward fundamental solution of the wave operator.

First order pseudodifferential operators on the torus: Normal forms, diophantine phenomena and global hypoellipticity

We show global normal forms of first order real principal type C∞ and Gevrey pseudodifferential operators on the torus T2 under generalized small divisor conditions on the principal symbol. We introduce a family of logarithmic Hausdorff dimensions associated to the Gevrey classes in order to examine the corresponding exceptional sets. We study the global hypoellipticity of such operators through inhomogenous Diophantine conditions for the normal forms.

On the propagation of a certain polarization set for semilinear systems of real principal type

On the propagation of a certain polarization set for semilinear systems of real principal type