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We address the open problem of determining which classes of time-dependentlinear Schr\"odinger equations and focusing and defocusing cubic and quinticnon-linear Schr\"odinger equations (NLS) on unbounded domains that can becomputed by an algorithm. We demonstrate how such an algorithm in general doesnot exist, yielding a substantial classification theory of which problems inquantum mechanics that can be computed. Moreover, we establish classificationson which problems that can be computed with a uniform bound on the runtime, asa function of the desired $\epsilon$-accuracy of the approximation. Thisinclude linear and nonlinear Schr\"odinger equations for which we providepositive and negative results and conditions on both the initial state and thepotentials such that there exist computational (recursive) a priori bounds thatallow reduction of the IVP on an unbounded domain to an IVP on a boundeddomain, yielding an algorithm that can produce an $\epsilon$-approximation. Inaddition, we show how no algorithm can decide, and in fact not verify norfalsify, if the focusing NLS will blow up in finite time or not, yet, for thedefocusing NLS, solutions can be computed given mild assumptions on the initialstate and the potentials. Finally, we show that solutions to discrete NLSequations (focusing and defocusing) on an unbounded domain can always becomputed with uniform bounds on the runtime of the algorithm. The algorithmspresented are not just of theoretical interest, but efficient and easy toimplement in applications. Our results have implications beyond computationalquantum 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 problemsmay be solved by computer assisted proofs.

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