On the Schrödinger and Carroll Schrödinger Equations: Dualities and Applications

TL;DR

The paper builds a unified operator framework that links the Schrödinger and Carroll-Schroedinger equations in $1{+}1$ dimensions, identifying a precise shared-solution criterion via $[\hat{\mathcal{H}},\hat{\mathcal{F}}]=0$ and deriving a potential map between $V_{sch}$ and $V_{car}$ through a Schwarzian relation. It then constructs a potential-dependent reparametrization $x=\delta(t)$ to transform space- and time-local problems into dual Schrödinger problems, relates conserved densities and currents through a gauge transform followed by a coordinate inversion, and presents an equal-$x$ Hilbert-space formulation with unitary $x$-evolution. The work also explores dynamical consequences including a Carrollian dispersion relation via ultra-boost, Hamilton–Jacobi classical limits, exact Gaussian-packet solutions, and a Dyson expansion for general $V(x,t)$, providing a practical dictionary for translating between Schrödinger and Carrollian dynamics. Collectively, these results lay a foundation for extending Carrollian quantum dynamics to higher dimensions and many-body settings, with potential implications for ultra-relativistic regimes, holography, and beyond.

Abstract

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 $L^2(R_t)$, 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.