Renormalization group invariance of quantum mechanics

We propose a framework to renormalize the nonrelativistic quantum mechanics with arbitrary singular interactions. The scattering equation is written to have one or more subtraction in the kernel at a given energy scale. The scattering amplitude is the solution of a nth order derivative equation in respect to the renormalization scale, which is the nonrelativistic counterpart of the Callan–Symanzik formalism. Scaled running potentials for the subtracted equations keep the physics invariant for a sliding subtraction point. An example of a singular potential, that requires more than one subtraction to renormalize the theory is shown.