Non-trapping Condition Research Articles

Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary

We study the quasilinear Maxwell system with a strictly positive, state dependent boundary conductivity. For small data we show that the solution exists for all times and decays exponentially to 0. As in related literature we assume a nontrapping condition. Our approach relies on a new trace estimate for the corresponding non-autonomous linear problem, an observability-type estimate, and a detailed regularity analysis. The results are improved in the linear autonomous case, using properties of the Helmholtz decomposition in Sobolev spaces of (small) negative order.

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.

Hybridization of the rigorous coupled-wave approach with transformation optics for electromagnetic scattering by a surface-relief grating

We hybridized the rigorous coupled-wave approach (RCWA) with transformation optics to develop a hybrid coordinate-transform method for solving the time-harmonic Maxwell equations in a 2D domain containing a surface-relief grating. In order to prove that this method converges for the p-polarization state, we studied several different but related scattering problems. The imposition of generalized non-trapping conditions allowed us to prove a-priori estimates for these problems. To do this, we proved a Rellich identity and used density arguments to extend the estimates to more general problems. These a-priori estimates were then used to analyze the hybrid method. We obtained convergence rates with respect to two different parameters, the first being a slice thickness indicative of spatial discretization in the depth dimension, the second being the number of terms retained in the Rayleigh–Bloch expansions of the electric and magnetic field phasors with respect to the other dimension. Testing with a numerical example revealed faster convergence than our analysis predicted. The hybrid method does not suffer from the Gibbs phenomenon seen with the standard RCWA.

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.

Global Strichartz estimates for an inhomogeneous Maxwell system

We show global-in-time Strichartz estimates for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as and a non-trapping condition. The proof is based on smoothing estimates in weighted L 2 spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem.

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.

Global in time Strichartz estimates for the fractional Schrödinger equations on asymptotically Euclidean manifolds

In this paper, we prove global in time Strichartz estimates for the fractional Schrödinger operators, namely e−itΛgσ with σ∈(0,∞)\\{1} and Λg:=−Δg where Δg is the Laplace–Beltrami operator on asymptotically Euclidean manifolds (Rd,g). Let f0∈C0∞(R) be a smooth cutoff equal 1 near zero. We firstly show that the high frequency part (1−f0)(P)e−itΛgσ satisfies global in time Strichartz estimates as on Rd of dimension d≥2 inside a compact set under non-trapping condition. On the other hand, under the moderate trapping assumption (1.12), the high frequency part also satisfies the global in time Strichartz estimates outside a compact set. We next prove that the low frequency part f0(P)e−itΛgσ satisfies global in time Strichartz estimates as on Rd of dimension d≥3 without using any geometric assumption on g. As a byproduct, we prove global in time Strichartz estimates for the fractional Schrödinger and wave equations on (Rd,g), d≥3 under non-trapping condition.

Inverse source problems without (pseudo) convexity assumptions

We study the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple wave numbers. The main goal of this paper is to study the uniqueness and increasing stability when the (pseudo)convexity or non-trapping conditions for the related hyperbolic problem are not satisfied. We consider general elliptic equations of the second order and arbitrary observation sites. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. Numerical examples in 2 spatial dimension support the analysis and indicate the increasing stability for large intervals of the wave numbers, while analytic proofs of the increasing stability are not available.

Multi-Center Vector Field Methods for Wave Equations

We develop the method of vector-fields to further study Dispersive Wave Equations. Radial vector fields are used to get a-priori estimates such as the Morawetz estimate on solutions of Dispersive Wave Equations. A key to such estimates is the repulsiveness or nontrapping conditions on the flow corresponding to the wave equation. Thus this method is limited to potential perturbations which are repulsive, that is the radial derivative pointing away from the origin. In this work, we generalize this method to include potentials which are repulsive relative to a line in space (in three or higher dimensions), among other cases. This method is based on constructing multi-centered vector fields as multipliers, cancellation lemmas and energy localization.

Variational approach to scattering by unbounded rough surfaces with Neumann and generalized impedance boundary conditions

This paper is concerned with problems of scattering of time-harmonic electromagnetic and acoustic waves from an innite penetrable medium with a certain non-trapping and geometric conditions.

Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials

The present paper is concerned with Schr\"odinger equations with variable coefficients and unbounded electromagnetic potentials, where the kinetic energy part is a long-range perturbation of the flat Laplacian and the electric (resp. magnetic) potential can grow subquadratically (resp. sublinearly) at spatial infinity. We prove sharp (local-in-time) Strichartz estimates, outside a large compact ball centered at origin, for any admissible pair including the endpoint. Under the nontrapping condition on the Hamilton flow generated by the kinetic energy, global-in-space estimates are also studied. Finally, under the nontrapping condition, we prove Strichartz estimates with an arbitrarily small derivative loss without asymptotic flatness on the coefficients.

Strichartz estimates for Schrödinger equations with variable coefficients and potentials at most linear at spatial infinity

In the present paper we consider Schrodinger equations with variable coefficients and potentials, where the principal part is a long-range perturbation of the flat Laplacian and potentials have at most linear growth at spatial infinity. We then prove local-in-time Strichartz estimates, outside a large compact set centered at origin, without loss of derivatives. Moreover we also prove global-in-space Strichartz estimates under the non-trapping condition on the Hamilton flow generated by the kinetic energy.

Strichartz Estimates for Schrödinger Equations on Scattering Manifolds

The present article is concerned with Schrödinger equations on non-compact Riemannian manifolds with asymptotically conic ends. It is shown that, for any admissible pair (including the endpoint), local in time Strichartz estimates outside a large compact set are centered at origin hold. Moreover, we prove global in space Strichartz estimates under the nontrapping condition on the metric.

Local well-posedness of a dispersive Navier-Stokes system

A. We establish local well-posedness and smoothing results for the Cauchy problem of a degenerate dispersive Navier-Stokes system that arises from kinetic theory. Under assumptions that the initial data satisfy asymptotic flatness and nontrapping conditions, we show there exists a unique classical solution for a finite time. Due to degeneracies in both dissipation and dispersion for the system, different components of the solution gain different regularity. Couplings of these components are analyzed using pseudodifferential operators.

Electromagnetic Wave Scattering from Rough Penetrable Layers

We consider scattering of time-harmonic electromagnetic waves from an unbounded penetrable dielectric layer mounted on a perfectly conducting infinite plate. This model describes, for instance, propagation of monochromatic light through dielectric photonic assemblies mounted on a metal plate. We give a variational formulation for the electromagnetic scattering problem in a suitable Sobolev space of functions defined in an unbounded domain containing the dielectric structure. Further, we derive a Rellich identity for a solution to the variational formulation. For simple material configurations and under suitable nontrapping and smoothness conditions, this integral identity allows us to prove an a priori estimate for such a solution. A priori estimates for solutions to more complicated material configurations are then shown using a perturbation approach. While the estimates derived from the Rellich identity show that the electromagnetic rough surface scattering problem has at most one solution, a limiting absorption argument finally implies existence of a solution to the problem.

Endpoint Strichartz estimates for the magnetic Schrödinger equation

We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n ⩾ 3 . The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the potentials, while the potentials themselves can be large. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials.

Molecular scattering and Born-Oppenheimer approximation

In this paper, we study the scattering wave operators for diatomic molecules by using the Born� Oppenheimer approximation. Assuming that the ratio h2 between the electronic and nuclear masses is small, we construct adiabatic wave operators that, under some non-trapping conditions, approximate the two-cluster wave operators up to any power of the parameter h.

Singularities of solutions to the Schrödinger equation on scattering manifold

In this paper we study microlocal singularities of solutions to Schr\"odinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in terms of the initial condition and the classical scattering maps under the nontrapping condition. Our result is closely related to a recent work by Hassell and Wunsch, though our model is more general and the method, which relies heavily on scattering theoretical ideas, is simple and quite different. In particular, we use an Egorov-type argument in the standard pseudodifferential symbol classes, and avoid using Legendre distributions. In the proof, we employ a microlocal smoothing property in terms of the {\it radially homogenous wave front set}, which is more precise than the preceding results.

A variational method for wave scattering from penetrable rough layers

We consider time-harmonic wave scattering from unbounded penetrable rough layers and provide existence theory for this problem via variational formulations. A rough layer is a model for a stratified and unbounded inhomogeneous medium where the refractive index varies. For our variational approach, the refractive index can be real or (partially) complex valued and is allowed to jump across interfaces. However, the index needs to satisfy a non-trapping condition, which requires, roughly speaking, monotonicity in the direction normal to the layer. In the half-space above and below the rough layer, we set up a radiation condition using the angular spectrum representation. Due to the unbounded setting, we establish integral formulas similar to Rellich's identity to obtain an a priori bound for a variational solution of the rough-layer scattering problem. This a priori bound is the basis for our existence result. We further provide regularity theory for the rough-layer scattering problem and give bounds on its frequency dependence. Our existence results hold for all wave numbers and apply to acoustic scattering in dimension two and three as well as to electromagnetic scattering for the electric and the magnetic mode.

The non-trapping degree of scattering

We consider classical potential scattering. If at energy E no orbit is trapped, the Hamiltonian dynamics define an integer-valued topological degree deg(E) ⩽ 1. This is calculated explicitly for all potentials, and exactly integers ⩽1 are shown to occur for suitable potentials.The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that boundary of Hill's region in configuration space is either empty or homeomorphic to a sphere.However, in many situations one can decompose a potential into a sum of non-trapping potentials with a non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering.