On the Schrödinger and Carroll Schrödinger Equations: Dualities and Applications
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| Format: | Preprint |
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2025
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| author | Rojas, José Casanova, Enrique Arias, Melvin |
| author_facet | Rojas, José Casanova, Enrique Arias, Melvin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_21597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Schrödinger and Carroll Schrödinger Equations: Dualities and Applications Rojas, José Casanova, Enrique Arias, Melvin Quantum Physics 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. |
| title | On the Schrödinger and Carroll Schrödinger Equations: Dualities and Applications |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2510.21597 |