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Recently, the formalism needed to relate the finite-volume spectrum ofsystems of nondegenerate spinless particles has been derived. In this work wediscuss a range of issues that arise when implementing this formalism inpractice, provide further theoretical results that can be used to check theimplementation, and make available codes for implementing the three-particlequantization condition. Specifically, we discuss the need to modify the upperlimit of the cutoff function due to the fact that the left-hand cut in thescattering amplitudes for two nondegenerate particles moves closer tothreshold; we describe the decomposition of the three-particle amplitude$\mathcal{K}_\text{df,3}$ into the matrix basis used in the quantizationcondition, including both $s$ and $p$ waves, with the latter arising in theamplitude for two nondegenerate particles; we derive the threshold expansionfor the lightest three-particle state in the rest frame up to$\mathcal{O}(1/L^5)$; and we calculate the leading-order predictions in chiralperturbation theory for $\mathcal{K}_\text{df,3}$ in the $\pi^+\pi^+K^+$ and$\pi^+K^+K^+$ systems. We focus mainly on systems with two identical particlesplus a third that is different ("2+1" systems). We describe the formalism infull detail, and present numerical explorations in toy models, in particularchecking that the results agree with the threshold expansion, and making aprediction for the spectrum of $\pi^+\pi^+K^+$ levels using the two- andthree-particle interactions predicted by chiral perturbation theory.

Implementing the three-particle quantization condition for $\pi^+\pi^+K^+$ and related systems