  We demonstrate how to apply the tensor-train format to solve the
time-independent Schr\"{o}dinger equation for quasi one-dimensional excitonic
chain systems with and without periodic boundary conditions. The coupled
excitons and phonons are modeled by Frenkel-Holstein type Hamiltonians with
on-site and nearest-neighbor interactions only. We reduce the memory
consumption as well as the computational costs significantly by employing
efficient decompositions to construct low rank tensor-train representations,
thus mitigating the curse of dimensionality. In order to compute also higher
quantum states, we introduce an approach which directly incorporates the
Wielandt deflation technique into the alternating linear scheme for the
solution of eigenproblems. Besides systems with coupled excitons and phonons,
we also investigate uncoupled problems for which (semi-)analytical results
exist. There, we find that in case of homogeneous systems the tensor-train
ranks of state vectors only marginally depend on the chain length which results
in a linear growth of the storage consumption. However, the CPU time increases
slightly faster with the chain length than the storage consumption because the
alternating linear scheme adopted in our work requires more iterations to
achieve convergence for longer chains and a given rank. Finally, we demonstrate
that the tensor-train approach to the quantum treatment of coupled excitons and
phonons makes it possible to directly tackle the phenomenon of mutual
self-trapping. We are able to confirm the main results of the Davydov theory,
i.e., the dependence of the wavepacket width and the corresponding
stabilization energy on the exciton-phonon coupling strength, though only for a
certain range of that parameter. In future work, our approach will allow
calculations also beyond the restrictions of the Frenkel-Holstein type
Hamiltonians.

Computational Physics;Quantum Physics