Non-trapping Condition Research Articles

Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed

The paper contains an analytic reconstruction formula in thermoacoustic and photoacoustic tomography. It works for any geometry of point detectors placement along a closed surface and for variable sound speed satisfying a non-trapping condition. It is shown how this formula leads in particular to eigenfunction expansion reconstructions, including those recently obtained for the case of a uniform background. A uniqueness of reconstruction result is also obtained.

Besov estimates in the high-frequency Helmholtz equation, for a non-trapping and [formula omitted] potential

We study the high-frequency Helmholtz equation, for a potential having C 2 smoothness, and satisfying the non-trapping condition. We prove optimal Morrey–Campanato estimates that are both homogeneous in space and uniform in the frequency parameter. The homogeneity of the obtained bounds, together with the weak assumptions we require on the potential, constitute the main new result in the present text. Our result extends previous bounds obtained by Perthame and Vega, in that we do not assume the potential satisfies the virial condition, a strong form of non-trapping.

Carleman estimates for the non-stationary Lamé system and the application to an inverse problem

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lame system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lame coefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.

Smoothing effect for Schrödinger equations

On démontre que l'hypothèse de non capture est nécessaire pour l'effet régularisant ($H^{1/2}$) pour l'équation de Schrödinger avec conditions aux limites de Dirichlet à l'extérieur d'un domaine de $\mathbb{R}^d$. On donne aussi une classe d'obstacles captifs (l'exemple d'Ikawa) pour lesquels on démontre un effet régularisant affaibli ($H^{1/2 - \varepsilon}$).

Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles withat most two incoming waves

We consider a two-dimensional inverse scattering problem of determining a sound-softobstacle from the far field pattern and, under a non-trapping condition, we establishuniqueness within a class of polygonal domains by means of a single incoming plane wave.Moreover, we will show similar uniqueness in the sound-hard case by means of twoincoming plane waves. The key is the analyticity of the solution of the scattering problemand the reflection of solutions.

STRICHARTZ ESTIMATES FOR A SCHRÖDINGER OPERATOR WITH NONSMOOTH COEFFICIENTS

ABSTRACT We prove Strichartz type estimates for the Schrödinger equation corresponding to a second order elliptic operator with variable coefficients. We assume that the coefficients are a C 2 compactly supported perturbation of the identity, satisfying a nontrapping condition.

Remarks on Carleman estimates and exact controllability of the Lamé system

In this paper we established the Carleman estimate for the two dimensional Lamé system with the zero Dirichlet boundary conditions. Using this estimate we proved the exact controllability result for the Lamé system with with a control locally distributed over a subdomain which satisfies to a certain type of nontrapping conditions.

The spectral asymptotics of elliptic operators of Schr�dinger type on a hyperbolic space

For self-adjoint second-order elliptic differential operators that satisfy the non-trapping condition on the n-dimensional hyperbolic spaceHnand coincide with the operator\( - \Delta - \left( {\frac{{n - 1}}{2}} \right)^2 \)in a neighborhood of infinity, where Δis the Laplace-Beltrami operator onHn,we obtain the complete asymptotic expansion of the spectral function as λ → +∞.For self-adjoint operators of the form (−Δ)\(\left( { - \Delta } \right)^{m/2} \)+Qm−r,where Qm−r is a pseudodifferential operator of order m−r that is automorphic with respect to a discrete group of isometries of the spaceHnwhose fundamental domain has finite volume, we introduce the spectral distribution function N(λ),which is the analog of the integrated state density, and we find its asymptotics up to order O(λ(n−r)/m)as λ → +∞.Bibliography: 49 titles.

Semiclassical resolvent estimates for N-body Schrödinger operators

In this paper we prove for a generalized N-body Schrödinger operator that the non-trapping condition on the classical hamiltonian and all classical subhamiltonians is both necessary and sufficient for obtaining good semiclassical bounds on the boundary values of the resolvent and their energy derivatives. We accomplish this by generalizing Gérard's geometrical construction of an escape function for three-body problems to N-body problems.

The Mourre estimate in the semiclassical limit

The Mourre estimate is proved in the semiclassical limit for Schrodinger operators satisfying a nontrapping condition. Moreover, an abstract form of a theorem concerning the absence of resonances is given.

A note on the absence of resonances for Schr�dinger operators

A simple proof of the absence of resonances for Schrodinger operators in the semiclassical limit is given under a generalized scaling and nontrapping (or virial) condition. The method of local distortion due to Hunziker is employed and a pseudo-differential operator calculus is used to construct the parametrix. The result is primarily intended for applications to the shape resonance, but can be also applied to various models including Stark effect and periodic potentials.

Time-decay of scattering solutions and resolvent estimates for semiclassical Schrödinger operators

We study the semiclassical Schrödinger operator, − h 2 Δ + V( x), h ϵ ]0, 1], and establish various results on the time-decay of scattering solutions and the semiclassical bounds for the powers of the resolvent. For short range potentials, the time-decay results obtained in this paper are the best possible and this enables us to conclude the equivalence between the non-trapping condition in classical mechanics and the uniform time-decay of wave functions in quantum mechanics.

On the location of resonances for Schrödinger operators in the semiclassical limit I. Resonances free domains

We show that under suitable non-trapping conditions on the potential V at energy E the Schrödinger operator H = − h ̵ 2 Δ + V exhibits no resonance in a neighbourhood of E for small h ̵ . We briefly discuss some critical values of E like thresholds or barrier tops.

On the mathematical theory of predissociation

We investigate spectral properties of molecular Schrodinger operators in the adiabatic aproximation. A mathematical theory of predissociation is presented in terms of the corresponding matrix Schrodinger operator and its resonances. Under a suitable nontrapping condition we give a convergent tunneling expansion for the location of resonances exponentially close to the real axis.

On the absence of resonances for Schr�dinger operators with non-trapping potentials in the classical limit

We provide bounds on resolvents of dilated Schrodinger operators via exterior scaling. This depends crucially on a non-trapping condition on the potential which has a clear interpretation in classical mechanics. These bounds imply absence of resonances due to the tail of the potential in the shape resonance problem.

Scattering amplitude and algorithm for solving the inverse scattering problem for a class of non-convex obstacles

Consider a class R 1 of non-convex smooth obstacles B such that there exist a number of directions { s j , − s j }, j = 1, 2,…, N, uniformly distributed on the unit sphere with the following property: the rays incident in the directions +- s j are reflected by the obstacle once according to geometrical optics laws. It is also assumed that the obstacles in R 1, satisfy a non-trapping condition. For B ϵ R 1 the high frequency asymptotics of the scattering amplitude f( s, s 0, k) is studied and analytical formulas are derived for determining the surface of the obstacle from the scattering amplitude known at a fixed large wave number k, and for the directions s 0 and s of the incident and scattered waves in solid angles around the directions ± s j , j = 1,…, N. The investigation of the scattering amplitude and the solution to the inverse scattering problem are based on approximate formulas for f derived from the Kirchhoff approximation, and these formulas are justified for non-convex obstacles of class R 1.