Limiting Absorption Principle Research Articles

Limiting absorption principle and radiation conditions for Schrödinger operators with long-range potentials

We show Rellich's theorem, the limiting absorption principle, and a Sommerfeld uniqueness result for a wide class of one-body Schrödinger operators with long-range potentials, extending and refining previously known results. Our general method is based on elementary commutator estimates, largely following the scheme developed recently by Ito and Skibsted.

On spectral and scattering theory for one-body quantum systems in crossed constant electric and magnetic fields

In this paper, we study the spectral and scattering theory on the Hamiltonians which govern one-body quantum systems in crossed constant electric and magnetic fields. As one of the spectral features of the Hamiltonians, we show the absence of bound states of theirs under some appropriate conditions on the potentials. Moreover, by using the limiting absorption principle for the Hamiltonians, we give a simplified proof of the asymptotic completeness in short-range scattering.

PLANE PROBLEMS ABOUT THE ACTION OF OSCILLATING LOAD ON THE BOUNDARY OF AN ELASTIC ISOTROPIC LAYER IN THE PRESENCE OF SURFACE STRESSES

In this paper, symmetric and antisymmetric plane problems about the action of oscillating load on the boundary of an elastic isotropic nanothin layer are considered. The nanoscale layer thickness is considered by introducing surface stresses in accordance with the Gurtin-Murdoch theory. According to this theory, it is assumed that, in addition to external loads, surface stresses act on the layer boundaries, which are described by Hooke's “surface” law. As a result, the properties of the elastic material of the layer with nanoscale thickness become different from the properties of the material of a regular-sized body, which is typical for nanomechanics problems. A standard technique was used for the solution of formulated problems, including the application of limiting absorption principle, the Fourier transform over infinitely extended coordinate and the theory of residues for finding the inverse Fourier transform. It is shown how it is possible to obtain solutions in the form of series in natural waves, in which the wave numbers are defined as the roots of the corresponding dispersion equations. For a specific example, dispersion relations were studied and graphs of the first dispersion curves were plotted. The behavior of barrier frequencies, changes in wave numbers and zones of existence of backward waves at different nanoscale layer thicknesses are analyzed. The results of the analysis showed that for an ultrathin layer, surface effects have a significant impact on the dispersion relations, and the trends in the dispersion curves can differ significantly for different modes and layer thicknesses.

A limiting absorption principle for high-order Schrödinger operators in critical spaces

A limiting absorption principle for high-order Schrödinger operators in critical spaces

Magnetic quantum currents in the presence of a Neumann wall

The Schrödinger operator with a constant magnetic field on a half-plane with Neumann boundary conditions is considered. Low energy currents flowing along the boundary are analyzed and used to establish a limiting absorption principle for the electrically perturbed operator.

Linear inviscid damping and vortex axisymmetrization via the vector field method

In this paper, we prove the inviscid damping and vortex axisymmetrization for the linearized Euler equation around the radially symmetric, strictly monotonic vorticity distribution via the vector field method. One of the key ingredients is to establish the limiting absorption principle for the Rayleigh equation. Compared with shear flows in a channel, the main difficulty comes from the degeneration of the pipe flow at the original point and infinity.

Spectral projectors, resolvent, and Fourier restriction on the hyperbolic space

We develop a unified approach to proving Lp−Lq boundedness of spectral projectors, the resolvent of the Laplace-Beltrami operator and its derivative on Hd. In the case of spectral projectors, and when p and q are in duality, the dependence of the implicit constant on p is shown to be sharp. We also give partial results on the question of Lp−Lq boundedness of the Fourier extension operator. As an application, we prove smoothing estimates for the free Schrödinger equation on Hd and a limiting absorption principle for the electromagnetic Schrödinger equation with small potentials.

On the scattering of a plane wave by a perturbed open periodic waveguide

We consider the scattering of a plane wave by a locally perturbed periodic (with respect to ) medium. If there is no perturbation, it is usually assumed that the scattered wave is quasi‐periodic with the same parameter as the incident plane wave. As it is well known, one can show existence under this condition but not necessarily uniqueness. Uniqueness fails for certain incident directions (if the wavenumber is kept fixed), and it is not clear which additional condition has to be assumed in this case. In this paper, we will analyze three concepts. For the limiting absorption principle (LAP), we replace the refractive index by in a layer of finite width and consider the limiting case . This will give an unsatisfactory condition. In a second approach, we require continuity of the field with respect to the incident direction. This will give the same satisfactory condition as the third approach where we approximate the incident plane wave by an incident point source and let the location of the source tend to infinity.

Dispersive equations on asymptotically conical manifolds: time decay in the low-frequency regime

On an asymptotically conical manifold, we prove time decay estimates for the flow of the Schrödinger wave and Klein–Gordon equations via some differentiability properties of the spectral measure. To keep the paper at a reasonable length, we limit ourselves to the low-energy part of the spectrum, which is the one that dictates the decay rates. With this paper, we extend sharp estimates that are known in the asymptotically flat case (see Bouclet and Burq in Duke Math J 170(11):2575–2629, 2021, https://doi.org/10.1215/00127094-2020-0080 ) to this more general geometric framework and therefore recover the same decay properties as in the Euclidean case. The first step is to prove some resolvent estimates via a limiting absorption principle. It is at this stage that the proof of the previously mentioned authors fails, in particular when we try to recover a low-frequency positive commutator estimate. Once the resolvent estimates are established, we derive regularity for the spectral measure that in turn is applied to obtain the decay of the flows.

Families of symmetries and the hydrogen atom

We study a new type of symmetry for the hydrogen atom involving algebraic families of groups parametrized by the energy value in the time-independent Schrödinger equation. We construct an algebraic family of Harish-Chandra modules from the solutions of the Schrödinger equation, and we characterize this family. We show that the subspaces of physical states may be obtained from our algebraic family using a Jantzen filtration, and we relate our algebraic methods with spectral theory and scattering theory using the limiting absorption principle.

A Mathematical Study of a Hyperbolic Metamaterial in Free Space

Wave propagation in hyperbolic metamaterials is described by the Maxwell equations with a frequency dependent tensor of dielectric permittivity, whose eigenvalues are of different signs. In this case the problem becomes hyperbolic (Klein-Gordon equation) for a certain range of frequencies. The principal theoretical and numerical difficulty comes from the fact that this hyperbolic equation is posed in a free space, without initial conditions provided. The subject of the work is the theoretical justification of this problem. In particular, this includes the construction of a radiation condition, a well-posedness result, a limiting absorption principle and regularity estimates on the solution.

Антиплоские задачи о движении осциллирующей нагрузки по границе упругой изотропной полосы при наличии поверхностных напряжений

In this paper, symmetric and antisymmetric antiplane problems about the motion with a constant subsonic velocity of an oscillating load along the boundary of an elastic isotropic nanothin strip are considered. The nanoscale strip thickness is considered by introducing surface stresses in accordance with the Gurtin-Murdoch theory. According to this theory, it is assumed that, in addition to external loads, surface stresses act on the layer boundaries, which are described by Hooke's surface law. As a result, the properties of the elastic material of the strip with nanoscale thickness become different from the material properties of a regular-sized body. A standard technique was used for the so-lution, including the application of limiting absorption principle, the Fourier transform over infinitely extended co-ordinate and the theory of residues for finding the inverse Fourier transform. For various strip thicknesses, solutions were obtained in the form of series in natural waves, dispersion relations were studied, and graphs of the displace-ment amplitudes along the thickness were plotted. The analysis showed that for fixed values of frequency and velocity of the source, the values of non-negative real wave numbers are greater in the presence of surface stresses than the values of the wave numbers for the classical case of the problem without surface stresses. It is noted that surface stresses have a significant effect only when the strip thickness decreases to nanosizes.

Maxwell's equations with hypersingularities at a conical plasmonic tip

In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical L2 framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behavior is selected via the limiting absorption principle.

Resolvent Estimates for Time-Harmonic Maxwell\u2019s Equations in the Partially Anisotropic Case

We prove resolvent estimates in L^p-spaces for time-harmonic Maxwell’s equations in two spatial dimensions and in three dimensions in the partially anisotropic case. In the two-dimensional case the estimates are sharp up to endpoints. We consider anisotropic permittivity and permeability, which are both taken to be time-independent and spatially homogeneous. For the proof we diagonalize time-harmonic Maxwell’s equations to equations involving Half-Laplacians. We apply these estimates to infer a Limiting Absorption Principle in intersections of L^p-spaces and to localize eigenvalues for perturbations by potentials.

A scattering problem for a local perturbation of an open periodic waveguide

In this paper, we consider the propagation of waves in an open waveguide in where the index of refraction is a local perturbation of a function which is periodic along the axis of the waveguide (which we choose to be the x1 axis) and equal to one for |x2| > h0 for some h0 > 0. Motivated by the limiting absorption principle (proven in an earlier paper by the author), we formulate a radiation condition which allows the existence of propagating modes and prove uniqueness, existence, and stability of a solution under the assumption that no bound states exist. In the second part, we determine the order of decay of the radiating part of the solution in the direction of the layer and in the direction orthogonal to it. Finally, we show that it satisfies the classical Sommerfeld radiation condition and allows the definition of a far field pattern.

Schrödinger operator with decreasing potential in a cylinder

The Schrödinger operator − Δ + V ( x , y ) -\Delta + V(x,y) is considered in a cylinder R m × U \mathbb {R}^m \times U , where U U is a bounded domain in R d \mathbb {R}^d . The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, | V ( x , y ) | ≤ C ⟨ x ⟩ − ρ |V(x,y)| \le C \langle x\rangle ^{-\rho } . If ρ > 1 \rho > 1 , then the wave operators exist and are complete; the Birman invariance principle and the limiting absorption principle hold true; the absolute continuous spectrum fills the semiaxis; the singular continuous spectrum is empty; the eigenvalues can accumulate to the thresholds only.

A Sharp Version of Price\u2019s Law for Wave Decay on Asymptotically Flat Spacetimes

We prove Price’s law with an explicit leading order term for solutions phi (t,x) of the scalar wave equation on a class of stationary asymptotically flat (3+1)-dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that phi (t,x)=c t^{-3}+{mathcal {O}}(t^{-4+}) for bounded |x|, where cin {mathbb {C}} is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise t^{-2 l-3} decay for waves with angular frequency at least l, and t^{-2 l-4} decay for waves which are in addition initially static. This definitively settles Price’s law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.

Limiting absorption principle and virtual levels of operators in Banach spaces

We review the concept of the limiting absorption principle and its connection to virtual levels of operators in Banach spaces.

A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations

In this work we investigate the Lp−Lq-mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we obtain an Lp−Lq-type Limiting Absorption Principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations. Moreover, we provide an improved version of the Limiting Absorption Principle for Hermitian (self-adjoint) Helmholtz systems.

Bands of pure absolutely continuous spectrum for lattice Schrödinger operators with a more general long range condition

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov–Vainberg Schrödinger operators, Δ + V and D + V on ℓ2(Zd), with emphasis on d = 1, 2, 3. Considered are electric potentials V satisfying a long range condition of the following type: V−τjκV decays appropriately at infinity for some κ∈N and all 1 ≤ j ≤ d, where τjκV is the potential shifted by κ units on the jth coordinate. More comprehensive results are obtained for small values of κ, e.g., κ = 1, 2, 3, 4. We work in a simplified framework in which the main takeaway appears to be the existence of bands where a limiting absorption principle holds, and hence, pure absolutely continuous spectrum exists. Other decay conditions at infinity for V arise from an isomorphism between Δ and D in dimension 2. Oscillating potentials are examples in application.