  We develop a computationally and numerically efficient method to calculate
binding energies and corresponding wave functions of quantum mechanical
three-body problems in low dimensions. Our approach exploits the tensor
structure of the multidimensional stationary Schr\"odinger equation, being
expressed as a discretized linear eigenvalue problem. In one spatial dimension,
we solve the three-body problem with the help of iterative methods. Here the
application of the Hamiltonian operator is represented by dense matrix-matrix
products. In combination with a newly-designed preconditioner for the
Jacobi-Davidson QR, our highly accurate tensor method offers a significantly
faster computation of three-body energies and bound states than other existing
approaches. For the two-dimensional case, we additionally make use of a hybrid
distributed/shared memory parallel implementation to calculate the
corresponding three-body energies. Our novel method is of high relevance for
the analysis of few-body systems and their universal behavior, which is only
governed by the particle masses, overall symmetries, and the spatial
dimensionality. Our results have straightforward applications for ultracold
atomic gases that are widespread and nowadays utilized in quantum sensors.
