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We explore a new numerical method for studying one-dimensional quantum systems in a trapping potential. We focus on the setup of an impurity in a fermionic background, where a single distinguishable particle interacts through a contact potential with a number of identical fermions. We can accurately describe this system, for various particle numbers, different trapping potentials and arbitrary finite repulsion, by constructing a truncated basis containing states at both zero and infinite repulsion. The results are compared with matrix product states methods and with the analytical result for two particles in a harmonic well. Copyright E. J. Lindgren et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 28-08-2019 Accepted 12-06-2020 Published 13-07-2020 Check for updates doi:10.21468/SciPostPhys.9.1.005

A systematic interpolatory method for an impurity in a one-dimensional fermionic background