# ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

@article{Aizawa2015oscillatorsFS,
title={ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras},
author={Naruhiko Aizawa and Zhanna Kuznetsova and Francesco Toppan},
journal={Journal of Mathematical Physics},
year={2015},
volume={56},
pages={031701}
}
• Published 31 December 2014
• Mathematics
• Journal of Mathematical Physics
We construct, for any given l=12+N0, the second-order, linear partial differential equations (PDEs) which are invariant under the centrally extended conformal Galilei algebra. At the given l, two invariant equations in one time and l+12 space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrodinger equation (recovered for l=12) in 1 + 1 dimension. The second equation (the “l-oscillator”) possesses a discrete, positive spectrum. It…

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## References

SHOWING 1-10 OF 93 REFERENCES

### Four types of (super)conformal mechanics: D-module reps and invariant actions

• Mathematics
• 2014
(Super)conformal mechanics in one dimension is induced by parabolic or hyperbolic/trigonometric transformations, either homogeneous (for a scaling dimension λ) or inhomogeneous (at λ = 0, with ρ an

### Intertwining operators for ℓ-conformal Galilei algebras and hierarchy of invariant equations

• Mathematics
• 2013
The ℓ-conformal Galilei algebra, denoted by gℓ(d),?> is a non-semisimple Lie algebra specified by a pair of parameters (d, ℓ). The algebra is regarded as a nonrelativistic analogue of the conformal

### On Schrödinger superalgebras

• Mathematics, Physics
• 1994
Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrodinger algebra (itself a conformal extension of the Galilei algebra) were

### Algebraic structure of Galilean superconformal symmetries

• Mathematics, Physics
• 2011
The semisimple part of d-dimensional Galilean conformal algebra g^(d) is given by h^(d)=O(2,1)+O(d), which after adding via semidirect sum the 3d-dimensional Abelian algebra t^(d) of translations,

### Super-Galilean conformal algebra in AdS/CFT

Galilean conformal algebra (GCA) is an Inonu–Wigner (IW) contraction of a conformal algebra, while Newton–Hooke string algebra is an IW contraction of an Anti-de Sitter (AdS) algebra, which is the

### Symmetries of the Schrödinger equation and algebra/superalgebra duality

Some key features of the symmetries of the Schrodinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the

### Schrodinger equations for higher order nonrelativistic particles and N-Galilean conformal symmetry

• Mathematics, Physics
• 2012
Department of Physics, Toho University, Miyama, Funabashi, 274-8510, Japan(Received 26 September 2011; published 15 February 2012)We consider Schro¨dinger equations for a nonrelativistic particle

### PT-symmetric quantum mechanics

• Physics
• 1999
This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex