论文标题
在数值算法的边缘上的无界域上的schrödinger方程的计算解决方案
Computing solutions of Schrödinger equations on unbounded domains- On the brink of numerical algorithms
论文作者
论文摘要
我们解决了开放的问题，即确定哪些时间依赖性的线性schrödinger方程，并在无限制的域上进行聚焦和散落的立方体和五季度非线性schrödinger方程（NLS），这些域可以通过算法计算。我们证明了这种算法通常不存在，从而产生了可以计算量子力学中哪些问题的实质性分类理论。此外，我们建立了可以通过在运行时与统一绑定的问题计算出的分类，这是近似值所需的$ε$ - 准确性的函数。其中包括线性和非线性schrödinger方程，我们为初始状态和电势提供了积极和负面的结果和条件，使得有计算（递归）的先验界限，允许将IVP降低到一个有限域上的IVP上的IVP，从而产生一个aLgorithm上的IVP，从而产生$ -Approximation $ -Approximatimatimapproximation。此外，我们还展示了如何无法决定算法，并且实际上没有验证或伪造算法，而焦点NLS是否会在有限的时间内爆炸，对于散落的NLS，可以在初始状态和电势上进行轻度假设，以计算解决方案。最后，我们表明，在算法的运行时始终可以在无界域上对离散域上离散的NLS方程（聚焦和散落）的解决方案进行计算。提出的算法不仅具有理论上的兴趣，而且在应用程序中易于实施。我们的结果超出了计算量子力学之外的含义，并且是解决计算数学基础上解决性复杂性指数（SCI）层次结构（SCI）层次结构的一部分。例如，我们的结果提供了分类，即通过计算机辅助证明可以解决哪些数学问题。
We address the open problem of determining which classes of time-dependent linear Schrödinger equations and focusing and defocusing cubic and quintic non-linear Schrödinger 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 $ε$-accuracy of the approximation. This include linear and nonlinear Schrödinger 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 $ε$-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.
