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

Abstract

We have extended a previous calculation of the energy of a weakly heterogeneous waveguide to fourth order in the density perturbation, deriving its general expression. For particular configurations where the second and third orders both vanish, we discover that the fourth order contribution lowers in general the energy of the state, below the threshold of the continuum. In these cases the waveguide possesses a localized state. We have applied our general formula to a solvable model with vanishing second and third orders reproducing the exact expression for the fourth order.

Bound States in Weakly Deformed Waveguides: Numerical Versus Analytical Results

“…The problem of calculating the emergence of trapped states in infinite slightly heterogeneous waveguides was recently considered by Amore et al [1,2], who obtained exact perturbative formulae that use the density inhomogeneity as a perturbation parameter. This approach extends a method previously developed by Gat and Rosenstein [13] for calculating the binding energy of threshold bound states.…”

“…In particular, Amore et al [2] gave a calculation up to third order, while Amore [1] extended the calculation to fourth order. These formulae have been tested on two exactly solvable models, reproducing the exact results up to fourth order.…”

“…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]).…”

“…The paper is organized as follows. In Section 2, we summarize the results of Amore et al [1,2]. In Section 3, we present a comparison of the results of different approaches to the three examples mentioned.…”