Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$

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 $\mathrm{T}\bar{\mathrm{T}}$ 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 $\mathrm{T}\bar{\mathrm{T}}$. 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.