Solving scattering problems in the half-line using methods developed for scattering in the full line

Abstract

We reduce the solution of the scattering problem defined on the half-line [0, ∞) by a real or complex potential v(x) and a general homogenous boundary condition at x = 0 to that of the extension of v(x) to the full line that vanishes for x < 0. We find an explicit expression for the reflection amplitude of the former problem in terms of the reflection and transmission amplitudes of the latter. We obtain a set of conditions on these amplitudes under which the potential in the half-line develops bound states, spectral singularities, and time-reversed spectral singularities where the potential acts as a perfect absorber. We examine the application of these results in the study of the scattering properties of a δ-function potential and a finite barrier potential defined in [0, ∞), discuss optical systems modeled by these potentials, and explore the configurations in which these systems act as a laser or perfect absorber. In particular, we derive an explicit formula for the laser threshold condition for a slab laser with a single mirror and establish the surprising fact that a nearly perfect mirror gives rise to a lower threshold gain than a perfect mirror. We also offer a nonlinear extension of our approach which allows for utilizing a recently developed nonlinear transfer matrix method in the full line to deal with finite-range nonlinear scattering interactions defined in the half-1 By being a scattering potential we mean that every solution of the Schrödinger equation (1) tends to a linear combination of plane waves as x → ∞. This is the case for potentials v(x) satisfying the Faddeev condition ∞ 0 (1 + x)|v(x)|dx < ∞, [4].

Low-frequency scattering defined by the Helmholtz equation in one dimension

“…Ref. [29] offers a simple method of extending the utility of the transfer matrix of potential scattering in the full line to the scattering problems in the half-line. Specifically, it relates the reflection amplitude R(k) of the latter to the entries of the transfer matrix for its trivial extension to the full line.…”

Dynamical formulation of low-energy scattering in one dimension

“…Ref. [31] outlines a simple mapping of the scattering problem given by ( 122) and (124) on the half-line to a scattering problem defined by the Schrödinger equation (1) for the potential, v(x) := V(x) for x ≥ 0, 0 for x < 0, (126) in the full line. This mapping allows for relating the reflection amplitude R(k) to the entries M ij (k) of the transfer matrix (alternatively the reflection and transmission amplitudes, R l/r (k) and T (k)) of the potential (126) according to…”

Transfer matrix for long-range potentials

“…This relation together with (4) and (27) allow us to identify M with U (∞, −∞), where U (x, x 0 ) is the evolution operator associated with the Hamiltonian H (x) and the initial 'time' x 0 , [42]. 1 Because we can express U (x, x 0 ) as the time-ordered exponential of H (x), i.e.,…”

“…In the latter case, it is σ 3 -pseudo-Hermitian [45], i.e., H (x) [45]. Because ψ 1 and ψ 2 are solutions of the Schrödinger equation ( 2), the corresponding two-component wave functions, Ψ 1 and Ψ 2 , solve (27). In light of (18), this implies that…”