On the Schrödinger and Carroll Schrödinger Equations: Dualities and Applications

We investigate precise structural relations between the standard Schrödinger equation and its Carrollian analogue-the Carroll-Schrödinger equation-in 1+1 dimensions, with emphasis on dualities, potential maps, and solution behavior. Our contributions proceed in the order of the paper: (i) we encode both dynamics with operators H and F under external potentials and explore conditions for obtaining the same type of solutions within both formalisms; (ii) we construct a potential-dependent reparametrization x = δ(t) mapping the space-independent Carroll equation to the time-independent Schrödinger equation, and derive a Schwarzian relation that specifies the map δ for any static V_sch (with harmonic, Coulomb-like, and free examples); (iii) we relate conserved densities and currents by removing V_car through a gauge transform followed by a coordinate inversion, establishing equivalence of the continuity equations; (iv) we obtain a Carrollian dispersion relation from an ultra-boost of the energy-momentum two-vector and also derive the classical limit of the Carroll wave equation via the Hamilton-Jacobi formalism; (v) we place Carroll dynamics on an equal-x Hilbert space L²\(R_t\), prove unitary x-evolution, and illustrate dynamics with an exactly solvable Gaussian packet and finite-time quantization for time-localized perturbations; and (vi) for general V(x; t) we perform a gauge reduction to an interaction momentum and set up a controlled Dyson expansion about solvable time profiles.