Restricting linear Schrödinger evolution to invariant families such as Gaussian and coherent states yields nonlinear, sometimes linear, reduced dynamics; a pulled-back Lagrangian formalism extends the construction to non-invariant families as an approximation.
Solving the Schrodinger Equation by Reduction to a First-order Differential Operator through a Coherent States Transform
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abstract
The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems.
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2019 1verdicts
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Nonlinear Dynamics from Linear Quantum Evolutions
Restricting linear Schrödinger evolution to invariant families such as Gaussian and coherent states yields nonlinear, sometimes linear, reduced dynamics; a pulled-back Lagrangian formalism extends the construction to non-invariant families as an approximation.