Limiting Absorption Principle Research Articles

Limiting absorption principles for the Navier equation in elasticity

We prove some a priori estimates for the resolvent of Navier equation in elasticity, when one approaches the spectrum (Limiting Absorption Principles). They are extensions of analogous estimates for the resolvent of the euclidean Laplacian in Rn. As a consequence, we get some results for the evolution equation and for the spectral measure

Time-decay of semigroups generated by dissipative Schrödinger operators

We establish a representation formula for semigroups of contractions in terms of a global limiting absorption principle. As applications, we prove long-time asymptotics and dispersive estimate of the semigroup generated by −iH where H is a dissipative Schrödinger operator.

Energy concentration and explicit Sommerfeld radiation condition for the electromagnetic Helmholtz equation

We study the electromagnetic Helmholtz equation(∇+ib(x))2u(x)+n(x)u(x)=f(x),x∈Rd, with the magnetic vector potential b(x) and n(x) a variable index of refraction that does not necessarily converge to a constant at infinity, but can have an angular dependence like n(x)→n∞(x|x|) as |x|→∞. We prove an explicit Sommerfeld radiation condition∫Rd|∇bu−in∞1/2x|x|u|2dx1+|x|<+∞ for solutions obtained from the limiting absorption principle and we also give a new energy estimate∫Rd|∇ωn∞(x|x|)|2|u|21+|x|dx<+∞, which explains the main physical effect of the angular dependence of n at infinity and deduces that the energy concentrates in the directions given by the critical points of the potential.

On the Linear Water Wave Problem in the Presence of a Critically Submerged Body

We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a critically submerged body (i.e., the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of a solution which satisfies the radiation conditions at infinity as well as at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyze its properties. Under a geometric condition introduced by V. Maz'ya in 1978, we prove an important property of the scattering matrix, which may be interpreted as the absence of total internal reflection. This property also allows us to obtain uniqueness and existence of a solution in some function spaces (e.g., $H^2_{loc}\cap L^\infty$) without use of the radiation conditions and the limiting absorption principle, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. The fact that the existence and uniqueness result does not rely on either the radiation conditions or the limiting absorption principle is the first result of this type known to us in the theory of linear wave problems in unbounded domains.

Limiting Absorption Principle and Perfectly Matched Layer Method for Dirichlet Laplacians in Quasi-cylindrical Domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.

Local Decay in Non-Relativistic QED

We prove the limiting absorption principle for a dressed electron at a fixed total momentum in the standard model of non-relativistic quantum electrodynamics. Our proof is based on an application of the smooth Feshbach-Schur map in conjunction with Mourre’s theory.

The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds

We describe the spectrum of the k k -form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the k k and k − 1 k-1 de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case. In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials V V such that the Schrödinger operator has compact resolvent, although in most directions the potential V V tends to − ∞ -\infty . We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension 4 whose cusps are rational homology spheres.

Uniform resolvent estimates for a non-dissipative Helmholtz equation

We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures of the solution when the source term concentrates on a bounded submanifold of R^n.

SPECTRAL RENORMALIZATION GROUP AND LOCAL DECAY IN THE STANDARD MODEL OF NON-RELATIVISTIC QUANTUM ELECTRODYNAMICS

We prove a limiting absorption principle for the standard model of non-relativistic quantum electrodynamics (QED) and for Nelson's model describing interactions of electrons with phonons. To this end, we use the spectral renormalization group technique on the continuous spectrum in conjunction with Mourre theory.

Dispersive Properties for Discrete Schrödinger Equations

In this paper we prove dispersive estimates for the system formed by two coupled discrete Schrödinger equations. We obtain estimates for the resolvent of the discrete operator and prove that it satisfies the limiting absorption principle. The decay of the solutions is proved by using classical and some new results on oscillatory integrals.

Perfectly Matched Layers for Diffraction Gratings in Inhomogeneous Media. Stability and Error Estimates

We study the scattering of acoustic or electromagnetic waves at diffraction gratings in inhomogeneous media. The refractive index stabilizes to 1 as the distance to the grating increases. Outgoing solutions are characterized by means of the limiting absorption principle. We prove the unique solvability of the problem with perfectly matched layer of finite length. Further, we show that solutions of the latter problem approximate outgoing solutions of the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity. Contribution of eigenvalues and resonances to the error of approximation is clarified.

The difference method in a diffraction problem: The technique of interior boundary condition

The diffraction problem for a plane wave on a half-plane covered by thin layer with an interface is solved by the difference method. The system of difference equations is derived from the variational principle. A boundary solution at infinity must be imposed; this is a radiation condition, which is used in the form of the limit absorption principle. The arising infinite system of difference equations is reduced to a finite part of the boundary (the interface) by using the technique of so-called interior boundary conditions in the sense of Ryaben’kii. The real conditions are found by the Fourier method with respect to one spatial variable in the form of Fourier or Laurent series in the corresponding variable, which converge either inside, outside, or on the unit circle. Above the upper boundary of the layer, all unknowns are eliminated by using the so-called grid Green function, that is, the resolving function for the half-plane satisfying the radiation condition at infinity. For the unknowns on the upper boundary of the layer, an equation in terms of a function of a complex variable of Wiener-Hopf type is obtained, which is solved by factorization. Factorization is performed numerically: the logarithm of the function is expanded in a bi-infinite series, which is replaced by a discrete Fourier series. The closing system in a neighborhood of the interface has order proportional to the number of points on the interface. Solving this system yields all of the required characteristics of the solution.

EIGENFUNCTION EXPANSIONS AND SPACETIME ESTIMATES FOR GENERATORS IN DIVERGENCE-FORM

Let [Formula: see text] be a formally self-adjoint (elliptic) operator in L2 (ℝn), n ≥ 2. The real coefficients aj, k(x) = ak, j(x) are assumed to be bounded and to coincide with -Δ outside of a ball. The paper deals with two topics: (i) An eigenfunction expansion theorem, proving in particular that H is unitarily equivalent to -Δ, and (ii) Global spacetime estimates for the associated inhomogeneous wave equation, proved under suitable ("nontrapping") additional assumptions on the coefficients. The main tool used here is a Limiting Absorption Principle (LAP) in the framework of weighted Sobolev spaces, which holds also at the threshold.

Limiting Absorption Principle for Some Long Range Perturbations of Dirac Systems at Threshold Energies

We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to study a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss some applications to the dispersive Helmholzt model in the quantum regime.

On the Green’s function for the Helmholtz operator in an impedance circular cylindrical waveguide

This paper addresses the problem of finding a series representation for the Green’s function of the Helmholtz operator in an infinite circular cylindrical waveguide with impedance boundary condition. Resorting to the Fourier transform, complex analysis techniques and the limiting absorption principle (when the undamped case is analyzed), a detailed deduction of the Green’s function is performed, generalizing the results available in the literature for the case of a complex impedance parameter. Procedures to obtain numerical values of the Green’s function are also developed in this article.

SPECTRAL SHIFT FUNCTION FOR OPERATORS WITH CROSSED MAGNETIC AND ELECTRIC FIELDS

We obtain a representation formula for the derivative of the spectral shift function ξ(λ; B, ∊) related to the operators [Formula: see text] and H(B, ∊) = H0(B, ∊) + V(x, y), B &gt; 0, ∊ &gt; 0. We establish a limiting absorption principle for H(B, ∊) and an estimate [Formula: see text] for ξ′(λ; B, ∊), provided λ ∉ σ(Q), where [Formula: see text].

Limiting absorption principle for the second quantization of self-adjoint operators

Limiting absorption principle for the second quantization of self-adjoint operators

Explicit Representation for the Infinite-Depth Two-Dimensional Free-Surface Green's Function in Linear Water-Wave Theory

In this paper we derive an explicit representation for the two-dimensional free-surface Green's function in water of infinite depth, based on a finite combination of complex-valued exponential integrals and elementary functions. This representation can easily and accurately be evaluated in a numerical manner, and its main advantage over other representations lies in its simplicity and in the fact that it can be extended towards the complementary half-plane in a straightforward manner. It seems that this extension has not been studied rigorously until now, and it is required when boundary integral equations are extended in the same way. For the computation of the Green's function, the limiting absorption principle and a partial Fourier transform along the free surface are used. Some of its properties are also discussed, and an expression for its far field is developed, which allows us to state appropriately the involved radiation condition. This Green's function is then used to solve the two-dimensional infinite-depth water-wave problem by developing a corresponding boundary integral equation, whose solution is determined by means of the boundary element method. To validate the computations, a benchmark problem based on a half-circle is presented and solved numerically.

Limiting absorption principle at low energies for a mathematical model of weak interaction: The decay of a boson

We study the spectral properties of a Hamiltonian describing the weak decay of spin 1 massive bosons into the full family of leptons. We prove that the considered Hamiltonian is self-adjoint, with a unique ground state and we derive a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold, for a sufficiently small coupling constant. As a corollary, we prove absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval. To cite this article: J.-M. Barbaroux, J.-C. Guillot, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Limiting Absorption Principle for the Dissipative Helmholtz Equation

Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the uniform resolvent estimate for the high-frequency Helmholtz equation when trapped classical trajectories meet the region where the absorption coefficient is non-zero. We also give the resolvent estimate in Besov spaces.