Rigorous analysis of the unified transform method and long-range instabilities for the inhomogeneous time-dependent Schrödinger equation on the quarter-plane

In this paper, we report on the discovery of a previously-unknown type of long-range instability phenomenon for the one-dimensional linear Schrodinger (LS) equation on the vacuum spacetime quarter-plane. More specifically, the inhomogeneous LS on the half-line, with generic initial data, boundary conditions and forcing term, is addressed, as an illustrative paradigm of our techniques, in a classical, smooth context via the formula proposed by the linear Fokas' unified transform method. We, first, present a new and suitable decomposition of that formula in the complex plane in order to appropriately interpret various terms appearing in the formula, thus securing convergence in a strictly defined sense. We also write the solution in a form consistent with the fundamental principle of Ehrenpreis and Palamodov. This novel analysis then allows for the necessary rigorous a posteriori verification of the full initial-boundary-value problem, for the first time. This is followed by a thorough investigation of the behavior of the solution near the boundaries of the spatiotemporal domain. We prove that the integrals in this representation converge uniformly to 'prescribed' values and the solution admits a smooth extension up to the boundary only if certain compatibility conditions are satisfied by the data (with direct implications for efficient numerics, well-posedness and control). Importantly, moreover, based on our analysis, we perform an effective asymptotic study of far-field dynamics. This leads to new explicit asymptotic formulae which characterize the properties of the solution in terms of (in)compatibilities of the data at the 'corner' of the quadrant. In particular, we found out that the asymptotic behavior of the solution is sensitive to perturbations of the data at the corner. In all cases, even assuming the initial data to belong to the Schwartz class, the solution loses this property as soon as time becomes positive (implying an infinite-speed type of singularity propagation). Hereby, the recent discovery of a novel type of a long-range instability effect for the Stokes equation is further corroborated and elucidated by revisiting a celebrated lower-order linear evolution partial differential equation (PDE). It thence transpires that our rigorous analytical approach is straightforwardly extendable to other Schrodinger-like evolution equations as well as more general problems with dispersion formulated on domains with a quasi-infinite boundary. Finally, although occurrence of the new instability is most stunning in the case discussed herein, it is naturally conjectured that analogous phenomena shall too appear in variable-coefficient and nonlinear settings which remain to be accordingly investigated.