The nonlinear Schrödinger equation on the half line

Abstract

Articles you may be interested inThe inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions J. Math. Phys. 55, 101505 (2014); 10.1063/1.4898768 Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditionsThe nonlinear Schrödinger equation on the half line with mixed boundary condition is investigated. After a brief introduction to the corresponding classical boundary value problem, the exact second quantized solution of the system is constructed. The construction is based on a new algebraic structure, which is called in what follows boundary algebra and which substitutes, in the presence of boundaries, the familiar Zamolodchikov-Faddeev algebra. The fundamental quantum field theory properties of the solution are established and discussed in detail. The relative scattering operator is derived in the Haag-Ruelle framework, suitably generalized to the case of broken translation invariance in space.

Exact Results for the One-Dimensional Many-Body Problem With Contact Interaction: Including a Tunable Impurity

“…The NLS model is one of most studied examples of integrable field theory for which a huge amount of exact results is known. In the same way, the Hamiltonian of [9] is the counterpart of the NLS model on the half-line whose symmetry is given by the reflection algebra [15], showing the consistency of the approach of [10]. In [15], the concept of boundary algebra [16] was crucial to establish all the properties of NLS on the half-line as an integrable system.…”

Factorization in integrable systems with impurity

“…This has been first presented in [17] and detailed in [21]. It is argued that solving (16) for T (k) in terms of R(k) and using the reflection equation, one gets that T (k) must satisfy the quartic relations (14) and (15) and not the cubic ones.…”

Solving the quantum nonlinear Schrödinger equation with δ-type impurity

“…It was argued in Ref. 32 that this solution is well-defined for a large class of functions ͓containing the Schwarz space S͑R͔͒ and an upper bound for g was given for the series ͑2.4͒ to converge uniformly in x. It also represents a physical field since it vanishes as x → ± ϱ.…”

“…We adopt the prescription detailed in Ref. 32 for the normal ordering denoted :¯: and apply it to ⌽ ␣ , ␣ = ±. Then following Ref.…”

“…We test this possibility on one of the most extensively studied integrable systems-the nonlinear Schrödinger ͑NLS͒ model. [25][26][27][28][29][30][31][32] More precisely, we are concerned below with the NLS model coupled to a delta-function impurity. The basic tool of our investigation is a specific exchange algebra, 6,7 called reflection-transmission ͑RT͒ algebra.…”