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We study the transition from integrability to chaos for the three-particleFermi-Pasta-Ulam- Tsingou (FPUT) model. We can show that both the quarticb-FPUT model ($\alpha$ = 0) and the cubic one ($\beta$ = 0) are integrable byintroducing an appropriate Fourier representation to express the nonlinearterms of the Hamiltonian. For generic values of $\alpha$ and $\beta$, the modelis non-integrable and displays a mixed phase space with both chaotic andregular trajectories. In the classical case, chaos is diagnosed by theinvestigation of Poincar\'e sections. In the quantum case, the level spacingstatistics in the energy basis belongs to the Gaussian orthogonal ensemble inthe chaotic regime, and crosses over to Poissonian behavior in thequasi-integrable low-energy limit. In the chaotic part of the spectrum, twogeneric observables obey the eigenstate thermalization hypothesis.

Quantization of integrable and chaotic three-particle Fermi-Pasta-Ulam-Tsingou models