  We show how local bounded interactions in an unbounded Hamiltonian lead to
eigenfunctions with favorable low-rank properties. To this end, we utilize
ideas from quantum entanglement of multi-particle spin systems. We begin by
analyzing the connection between entropy area laws and low-rank
approximability. The characterization for 1D chains such as Matrix Product
States (MPS) / Tensor Trains (TT) is rather extensive though incomplete. We
then show that a Nearest Neighbor Interaction (NNI) Hamiltonian has
eigenfunctions that are approximately separable in a certain sense. Under a
further assumption on the approximand, we show that this implies a constant
entropy bound. To the best of our knowledge, this work is the first analysis of
low-rank approximability for unbounded Hamiltonians. Moreover, it extends
previous results on entanglement entropy area laws to unbounded operators. The
assumptions include a variety of self-adjoint operators and have a physical
interpretation. The weak points are the aforementioned assumption on the
approximand and that the validity is limited to MPS/TT formats.
