Asymptotic behaviour of the spectrum of a waveguide with distant perturbations

Abstract

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator.

References in corpus (5)

• Bound states in weakly deformed strips and layers
• Exponential splitting of bound states in a waveguide with a pair of distant windows
• Lower bounds on the lowest spectral gap of singular potential Hamiltonians
• Distant perturbations of the Laplacian in a multi-dimensional space
• On local perturbations of Shrodinger operator in axis

Cited by in corpus (8)

• Tunneling resonances in systems without a classical trapping
• Bifurcations of thresholds in essential spectra of elliptic operators under localized non-Hermitian perturbations
• Discrete spectrum of thin PT-symmetric waveguide
• Expansion of the spectrum in the weak disorder regime for random operators in continuum space
• Quantum Hamiltonians with weak random abstract perturbation. II. Localization in the expanded spectrum
• Spectral localization for quantum Hamiltonians with weak random delta interaction
• On finitely many resonances emerging under distant perturbations in multi-dimensional cylinders
• Spectral gaps for self-adjoint second order operators