Computing solutions of Schr\"odinger equations on unbounded domains- On the brink of numerical algorithms

We address the open problem of determining which classes of time-dependent linear Schr\"odinger equations and focusing and defocusing cubic and quintic non-linear Schr\"odinger equations (NLS) on unbounded domains that can be computed by an algorithm. We demonstrate how such an algorithm in general does not exist, yielding a substantial classification theory of which problems in quantum mechanics that can be computed. Moreover, we establish classifications on which problems that can be computed with a uniform bound on the runtime, as a function of the desired $\epsilon$-accuracy of the approximation. This include linear and nonlinear Schr\"odinger equations for which we provide positive and negative results and conditions on both the initial state and the potentials such that there exist computational (recursive) a priori bounds that allow reduction of the IVP on an unbounded domain to an IVP on a bounded domain, yielding an algorithm that can produce an $\epsilon$-approximation. In addition, we show how no algorithm can decide, and in fact not verify nor falsify, if the focusing NLS will blow up in finite time or not, yet, for the defocusing NLS, solutions can be computed given mild assumptions on the initial state and the potentials. Finally, we show that solutions to discrete NLS equations (focusing and defocusing) on an unbounded domain can always be computed with uniform bounds on the runtime of the algorithm. The algorithms presented are not just of theoretical interest, but efficient and easy to implement in applications. Our results have implications beyond computational quantum mechanics and are a part of the Solvability Complexity Index (SCI) hierarchy and Smale's program on the foundations of computational mathematics. For example our results provide classifications of which mathematical problems may be solved by computer assisted proofs.