Deforming the ODE/IM correspondence with T T ¯

The ODE/IM correspondence is an exact link between classical and quantum integrable models. The primary purpose of this work is to show that it remains valid after TT perturbation on both sides of the correspondence. In particular, we prove that the deformed Lax pair of the sinh-Gordon model, obtained from the unperturbed one through a dynamical change of coordinates, leads to the same Burgers-type equation governing the quantum spectral flow induced by TT. Our main conclusions have general validity, as the analysis may be easily adapted to all the known ODE/IM examples involving integrable quantum field theories.