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.