  We propose a flexible power method for computing the leftmost, i.e.,
algebraically smallest, eigenvalue of an infinite dimensional tensor eigenvalue
problem, $H x = \lambda x$, where the infinite dimensional symmetric matrix $H$
exhibits a translational invariant structure. We assume the smallest eigenvalue
of $H$ is simple and apply a power iteration of $e^{-H}$ with the eigenvector
represented in a compact way as a translational invariant infinite Tensor Ring
(iTR). Hence, the infinite dimensional eigenvector can be represented by a
finite number of iTR cores of finite rank. In order to implement this power
iteration, we use a small parameter $t$ so that the infinite matrix-vector
operation $e^{-Ht}x$ can efficiently be approximated by the Lie product
formula, also known as Suzuki--Trotter splitting, and we employ a low rank
approximation through a truncated singular value decomposition on the iTR cores
in order to keep the cost of subsequent power iterations bounded. We also use
an efficient way for computing the iTR Rayleigh quotient and introduce a finite
size iTR residual which is used to monitor the convergence of the Rayleigh
quotient and to modify the timestep $t$. In this paper, we discuss 2 different
implementations of the flexible power algorithm and illustrate the automatic
timestep adaption approach for several numerical examples.
