On the Schrödinger and Carroll Schrödinger Equations: Dualities and Applications

• URL: http://arxiv.org/abs/2510.21597v1
• Date: Fri, 24 Oct 2025 16:02:08 GMT
• Title: On the Schrödinger and Carroll Schrödinger Equations: Dualities and Applications
• Authors: José Rojas, Enrique Casanova, Melvin Arias,
• Abstract summary: We investigate structural relations between the standard Schr"odinger equation and its Carrollian analogue-the Carroll-Schr"odinger equation-in 1+1 dimensions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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