Uniform Resolvent Research Articles

Components of topological uniform descent resolvent set and local spectral theory

In this paper, we shall study the components of topological uniform descent resolvent set ρud(T) for a bounded linear operator T acting on a Banach space. We first show the constancy of certain subspace valued mappings on the components of ρud(T). Then using these results and, the equivalences of SVEP at a point λ for T and T* in the case that λI-T has topological uniform descent, we obtain a classification of these components. Finally, we give some applications of the classification. In particular, we give a sufficient condition on an operator T such that its topological uniform descent spectrum σud(T) coincides with some distinguished parts of its spectrum σ(T).

Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

We consider a magnetic Schrödinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum.

Structural Resolvent Estimates and Derivative Nonlinear Schrödinger Equations

A refinement of uniform resolvent estimate is given and several smoothing estimates for Schrodinger equations in the critical case are induced from it. The relation between this resolvent estimate and radiation condition is discussed. As an application of critical smoothing estimates, we show a global existence results for derivative nonlinear Schrodinger equations.

Planar waveguide with “twisted” boundary conditions: Small width

We consider a planar waveguide with “twisted” boundary conditions. By twisting we mean a special combination of Dirichlet and Neumann boundary conditions. Assuming that the width of the waveguide goes to zero, we identify the effective (limiting) operator as the width of the waveguide tends to zero, establishes the uniform resolvent convergence in various possible operator norms, and gives the estimates for the rates of convergence. We show that studying the resolvent convergence can be treated as a certain threshold effect and we present an elegant technique which justifies such point of view.

Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows

We consider a planar waveguide modeled by the Laplacian in a straight infinite strip with the Dirichlet boundary condition on the upper boundary and with frequently alternating boundary conditions (Dirichlet and Neumann) on the lower boundary. The homogenized operator is the Laplacian subject to the Dirichlet boundary condition on the upper boundary and to the Dirichlet or Neumann condition on the lower one. We prove the uniform resolvent convergence for the perturbed operator in both cases and obtain the estimates for the rate of convergence. Moreover, we construct the leading terms of the asymptotic expansions for the first band functions and the complete asymptotic expansion for the bottom of the spectrum. Bibliography: 17 titles. Illustrations: 3 figures.

Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.

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.

On a waveguide with an infinite number of small windows

We consider a waveguide modeled by the Laplacian in a straight planar strip with the Dirichlet condition on the upper boundary, while on the lower one we impose periodically alternating boundary conditions with a small period. We study the case when the homogenization leads us to the Neumann boundary condition on the lower boundary. We establish the uniform resolvent convergence and provide the estimates for the rate of convergence. We construct the two-terms asymptotics for the first band functions of the perturbed operator and also the complete two-parametric asymptotic expansion for the bottom of its spectrum.

On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition

We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet–Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.

Uniform Resolvent Estimates for Magnetic Schrödinger Operators and Smoothing Effects for Related Evolution Equations

We prove uniform resolvent estimates for the magnetic Schrödinger operator in an exterior domain under smallness conditions on the magnetic fields and the scalar potential. The results are then used to obtain smoothing effects for the corresponding Schrödinger and Klein–Gordon evolution equations.

Homogenization of the planar waveguide with frequently alternating boundary conditions

We consider the Laplacian in a planar strip with a Dirichlet boundary condition on the upper boundary and with a frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary into small segments on which Dirichlet and Neumann conditions are imposed in turns. We show that under certain conditions the homogenized operator is the Dirichlet Laplacian and prove the uniform resolvent convergence. The spectrum of the perturbed operator consists of its essential part only and has a band structure. We construct the leading terms of the asymptotic expansions for the first band functions. We also construct the complete asymptotic expansion for the bottom of the spectrum.

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.

Approximation of a point perturbation on a Riemannian manifold

We show that the Hamiltonian of point interaction on a Riemannian manifold with bounded geometry can be obtained as a limit (in the sense of uniform resolvent convergence) of a sequence of scaling Hamiltonians with short-range interaction.

UNIFORM RESOLVENT CONVERGENCE OF LINEAR OPERATORS UNDER SINGULAR PERTURBATIONS

In this paper, we investigate sufficient conditions in order that a family T(ε)=T0+εT1 of closable linear operators with domain D(T(ε))=D(T0)∩​D(T1) converge to T0 as ε↓0 in the sense of uniform and strong resolvent convergence. The obtained abstract results are applied to selfadjoint and nonselfadjoint Schrödinger operators.

An elementary approach to Carleman-type resolvent estimates

A new elementary approach to uniform resolvent estimates of the Carleman-type is developed. Schatten-von Neumann's $$\mathfrak{S}_p $$ perturbations of self-adjoint and unitary operators are considered. Examples of typical growth are provided.

Spectral and scattering theory for 3-particle Hamiltonian with Stark effect : Non-existence of bound states and resolvent estimate

The present work is a continuation to [16], in which the author has proved the asymptotic completeness of wave operators for three-particle Stark Hamiltonians. In the proof there, the following two results about the spectral properties of two-particle subsystem Hamiltonians have played a central role: (1) non-existence of bound states; (2) uniform resolvent estimate at high energies. We here consider these two problems for three-particle systems and apply the obtained results to prove the asymptotic completeness for four-particle Stark Hamiltonians under the main assumption that any subsystem Hamiltonian does not have zero reduced charge.

Uniform resolvent convergence of linear operators under perturbations

Uniform resolvent convergence of linear operators under perturbations

Semiclassical resolvent estimates for two and three–body schrödinger operators

We prove uniform resolvent estimates for semiclassical three–body Schrodinger operators under a non–trapping condition for the classical flow of all subsystems. We also prove resolvent estimates for two–body Schrodinger operators with positive potentials when the energy level and the Planck constant tend both to zero.

Semiclassical resolvent estimates for two and three-body Schrödinger operators

We prove uniform resolvent estimates for semiclassical three–body Schrodinger operators under a non–trapping condition for the classical flow of all subsystems. We also prove resolvent estimates for two–body Schrodinger operators with positive potentials when the energy level and the Planck constant tend both to zero.