Abstract
The Newton–Hooke duality and its generalization to arbitrary power laws in classical, semiclassical and quantum mechanics are discussed. We pursue a view that the power-law duality is a symmetry of the action under a set of duality operations. The power dual symmetry is defined by invariance and reciprocity of the action in the form of Hamilton’s characteristic function. We find that the power-law duality is basically a classical notion and breaks down at the level of angular quantization. We propose an ad hoc procedure to preserve the dual symmetry in quantum mechanics. The energy-coupling exchange maps required as part of the duality operations that take one system to another lead to an energy formula that relates the new energy to the old energy. The transformation property of the Green function satisfying the radial Schrödinger equation yields a formula that relates the new Green function to the old one. The energy spectrum of the linear motion in a fractional power potential is semiclassically evaluated. We find a way to show the Coulomb–Hooke duality in the supersymmetric semiclassical action. We also study the confinement potential problem with the help of the dual structure of a two-term power potential.
Highlights
In recent years, numerous exoplanets have been discovered
The exchange operation E has a double role; it may decide the constant C appearing in the transformation r = Cρη, while it leads to an energy formula that relates the new energy to the old energy
(2) We have shown that the semiclassical action is symmetric under the set of duality operations {C, R, E, L} without T insofar as angular momentum L is treated as a continuous parameter, and observed that the power-duality is essentially a classical notion and breaks down at the level of angular quantization
Summary
Numerous exoplanets have been discovered. One of the best Doppler spectrographs to discover low-mass exoplanets using the radial velocity method are HARPS (High Accuracy Radial Velocity Planet Searcher) installed on ESO’s 3.6 m telescope at La Silla and ESPRESSO (Echelle Spectrograph for Rocky Exoplanet- and Stable Spectroscopic Observations) installed on ESO’s VLT at Paranal Observatory in Chile. As has been pointed out in the literature [10,18,22,23,24], the dual relation between the Kepler problem and the harmonic oscillator was already known in the time of Newton and Hooke. The path integral in the original formulation gives exact solutions only for quadratic systems including the harmonic oscillator, but fails in solving the hydrogen atom problem. Use of the KS transformation enables to convert the path integral for the hydrogen atom problem to that of the harmonic oscillator if the action of Feynman’s path integral is slightly modified with a fixed energy term. In the present paper we pursue the dual relation (power-duality) between two systems with arbitrary power-law potentials from the symmetry point of view. Appendix A presents the Newton–Hooke–Morse triality that relates the Newton–Hooke duality to the Morse oscillator
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