QM scattering amplitudes through the lens of homotopy transfer

We consider the 1D quantum scattering problem for a Hamiltonian with symmetries. We show that proper treatment of symmetries in the spirit of homological algebra leads to new objects, generalizing the well-known T- and K-matrices. Homological treatment implies that old objects and new ones are to be combined in a differential. This differential arises from homotopy transfer of induced interaction and symmetries on solutions of free equations of motion. Therefore, old and new objects satisfy remarkable quadratic equations. We construct an explicit example in SUSY QM on a circle to demonstrate the nontriviality of the above relation.