Weakly bound states in heterogeneous waveguides

Abstract

Abstract. We study the spectrum of the Helmholtz equation in a two-dimensional infinite waveguide, containing a weak heterogeneity localized at an internal point, and obeying Dirichlet boundary conditions at its border. We prove that, when the heterogeneity corresponds to a locally denser material, the lowest eigenvalue of the spectrum falls below the continuum threshold and a bound state appears, localized at the heterogeneity. We devise a rigorous perturbation scheme and derive the exact expression for the energy to third order in the heterogeneity.

Weakly Bound States in Heterogeneous Waveguides: A Calculation to Fourth Order

“…While it was already proved in Ref. [9] that E (3) 0 is finite for β → 0 + , as it can be checked explicitly using the results in Appendix B of Ref. [14], it is straightforward to verify that η 4a = η 4b = 0.…”

“…As we have discussed in Ref. [9], the identification of the unperturbed operator must be done with care, for the case of an infinite waveguide: as a matter of fact, the obvious candidate, corresponding to an infinite, straight and homogeneous waveguide cannot be used, since its spectrum is continuous and the fundamental mode can thus be excited to states which are arbitrarily close in energy. In this case, the perturbative formulae would contain infrared divergences, which would completely spoil the calculation.…”

“…[9], obtaining the exact general expression for the energy correction to fourth order in the density perturbation. The greater technical difficulty of the present (1351) P. Amore calculation derives both because from the larger number of terms and both from their different nature.…”

Bound States in Weakly Deformed Waveguides: Numerical Versus Analytical Results

“…The principal difficulty of the problems under consideration is that in each case the nonperturbed problem does not have eigenvalues, so that the standard regular perturbation theory is not applicable. Recently, yet another perturbative approach, which extends a method previously developed by Gat and Rosenstein [13], was proposed by Amore et al [1,2]. It has the advantage of using an auxiliary "unperturbed" problem, which does possess an eigenvalue and is exactly solvable, so that a standard perturbation procedure can be used to construct corrections up to any order.…”

“…Note that the approach of Amore et al [1,2] is equally applicable to weakly deformed or weakly heterogeneous waveguides. Therefore, we have chosen the following three examples: (a) an asymmetrically deformed waveguide (this is exactly the case considered by Bulla et al [8]); (b) an asymmetrically deformed waveguide with a localized heterogeneity (we are not aware of the existence of any results for this case in the literature); (c) a broken strip (investigated by different methods of Avishai et al [4] and Granot [14]).…”