On the Schr\"odinger and Carroll Schr\"odinger Equations: Dualities and Applications

TL;DR

This paper explores the deep structural relations, dualities, and applications of the Schr"odinger and Carroll-Schr"odinger equations in 1+1 dimensions, including solution mappings, conserved quantities, and quantum dynamics.

Contribution

It introduces a potential-dependent reparametrization linking Carroll and Schr"odinger equations, establishes dualities, and develops a framework for analyzing Carrollian quantum dynamics and solution behaviors.

Findings

Derived a Schwarzian relation for the solution map elta(t)

Established equivalence of continuity equations via gauge and coordinate transformations

Presented exact solutions and quantization methods for Carrollian quantum dynamics

Abstract

We investigate precise structural relations between the standard Schr\"odinger equation and its Carrollian analogue-the Carroll-Schr\"odinger 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 $x = \delta(t)$ mapping the space-independent Carroll equation to the time-independent Schr\"odinger equation, and derive a Schwarzian relation that specifies the map $\delta$ 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 $L^2(R_t)$, 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.