Hamiltonian deformations in quantum mechanics, TT¯, and the SYK model

Motivated by $T\overline{T}$, we introduce and study a wide class of solvable deformations of quantum-mechanical theories. These deformations map the Hamiltonian to a function of itself. We solve these theories by computing all finite-temperature correlation functions of the deformed theory in terms of the correlators of the undeformed theory. Applications to $\mathrm{AdS}/\mathrm{CFT}$, Sachdev-Ye-Kitaev, and the Schwarzian theory are considered. We write down the deformed Schwarzian action for an arbitrary Hamiltonian deformation and find that the maximal Lyapunov exponent is unchanged.