We examine a method for solving an infinite-dimensional tensor eigenvalue problem $H x = \lambda x$, where the infinite-dimensional symmetric matrix $H$ exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to $e^{-Ht}$ is used to obtain an approximation to the desired eigenvector. This infinite-dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step $t$ to ensure accurate and rapid convergence of the power method.

翻译：我们研究了一种求解无穷维张量特征值问题 $H x = \lambda x$ 的方法，其中无穷维对称矩阵 $H$ 具有平移不变结构。我们从数值线性代数的角度给出了这类问题的表述，并描述了如何将幂法应用于 $e^{-Ht}$ 以近似目标特征向量。该无穷维特征向量通过平移不变无穷张量环（iTR）以紧凑形式表示。利用低秩近似在保持近似特征向量 iTR 结构的同时，控制后续幂迭代的计算成本。我们展示了如何高效计算 iTR 特征向量近似的平均瑞利商，并引入了投影残差以监测其收敛性。在数值算例中，我们验证了该投影 iTR 残差的范数可用于自动调整时间步长 $t$，从而确保幂法准确且快速收敛。