We study the computational complexity of the eigenvalue problem for the Klein–Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein–Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein–Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.

中文翻译：

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计算克莱因-戈登谱


我们在可解复杂性指数层次结构的框架内研究克莱因-戈登方程特征值问题的计算复杂性。我们证明，具有线性衰减势的克莱因-戈登方程的特征值可以在单个极限内计算，并保证上面的误差范围。证明是建设性的，即我们获得了可以在计算机上实现的数值算法。此外，我们证明了克莱因-戈登方程点谱的抽象封装，并将我们的数值结果与这些封装进行了比较。最后，我们将实现的算法和抽象外壳应用于几个物理相关的势，例如 Sauter 和尖点势，并提供收敛和误差分析。