Moving boundary i.b.v.p.s for Schrödinger equations: solution bounds and related results
The paper is concerned with initial boundary value problems for Schrödinger equations in one spatial dimension, linear (\(iu_t=u_{xx}\)) and nonlinear (\(iu_t=u_{xx}-f(|u|^2)u\)), in a straight line region with prescribed, moving boundaries, upon which (time-dependent) Dirichlet conditions are specified. Bounds, in terms of data, are obtained for the \(L_{2}\) norm of the spatial derivative of the solutions, or for a measure related thereto: in the context of expanding boundaries, pointwise bounds for the solution may be inferred both in the linear case and in some nonlinear cases (e.g. the defocusing case). Asymptotic properties of the bounds for the aforementioned norm are discussed in the linear case. The methodology of the paper is based on a particular compact formula for the aforementioned norm of an arbitrary, complex-valued function whose values are assigned, as functions of time, on the assigned, moving boundaries of a straight line region.
scientific article; zbMATH DE number 5777330
scientific article