- The student is familar with Schrödinger equations on the real line, and its spectral properties includimg Jost solutions, reflection and transmission coefficients
- The student is knows what a Lax pair is, and how this can be used to solve certain integrable systems such as the KdV-equation using the inverse scattering method
- The student is familiar with the quantum inverse scattering method and Bethe Ansatz in the case of some other integrable systems
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We start with the study of Schrödinger equations on the real line with sufficiently fast decaying potential, which is viewed from an analytic point of view. In particular we study bound states -corresponding to the discrete spectrum- as well as the continuous spectrum. In more physical terms, we study the scattering data in terms of reflection and transmission coefficients obtained by studying special classes of solutions to the Schrödinger equation, known as the Jost solutions. Next we study the inverse problem, namely how to obtain the potential of a Schrödinger equation from the knowledge of the scattering data, i.e. the reflection and transmission coefficients, using the Gelfand-Levitan-Marchenko integral equation. We discuss scattering theory in general terms of deformation and approximation of operators on Hilbert spaces, and we in particular study the behaviour of the spectra of these operators under deformation.
Next the Korteweg-de Vries (KdV) equation is considered. This famous non-linear equation is an example of an integrable system, meaning that it has `sufficiently many' conserved quantities. The Lax representation of the KdV-equation is studied, and one of the operators in the Lax pair in the Lax representation is given by a time-dependent Schrödinger equation. It is shown how Schrödinger operators and their scattering data can be used to solve the KdV-equation, a method which is known as the ISM (Inverse Scattering Method). In particular, we show that the reflectionless case of the scattering data gives rise to the soliton solutions of the KdV-equation.
We plan to discuss other cases of integrable systems, namely Heisenberg spin chain models, and to study them in terms of QISM (quantum inverse scattering method) and the Bethe Ansatz. |
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