We present a method for solving eigenvalue problems on a quantum computer based on spectroscopy. The method works by coupling a "probe"qubit to a set of system simulation qubits and then time evolving both the probe and the system under Hamiltonian dynamics. In this way, we simulate spectroscopy on a quantum computer. We test our method on the IBM quantum hardware for a simple single-spin model and an interacting Kitaev chain model. For the Kitaev chain, we trace out the pseudotopological phase boundary for a two-site model.