Isochronous Oscillators

DOI

10.1142/S1402925110000611How to use a DOI?

Keywords

Nonlinear oscillators; isochronous Hamiltonians; quantization; equispaced spectrum; infinite degeneracy

Abstract

We exhibit the solution of the initial-value problem for the system of 2N + 2 oscillators characterized by the Hamiltonian

H(p^0,pˇ0,p^_,pˇ_,q^0,qˇ0,q^_,qˇ_)=12[p^02−pˇ02+Ω2(q^02−qˇ02)]        +q^0−Ωqˇ02b∑n=1N[p^n2−pˇn2+ωn2(q^n2−qˇn2)]+p^0−Ωq^0b∑n=1N[−p^npˇn+ωn2q^nqˇn]

where N is an arbitrary positive integer, Ω, b and ωn2 are N + 2 arbitrary real constants, q^m,qˇm with m = 0,1,…,N are the 2N + 2 canonical coordinates and p^m,pˇm the corresponding 2N + 2 canonical momenta. In the classical context the solution is completely periodic with period T = 2π/|Ω| (for arbitrary initial data). In the quantal context the (infinitely degenerate) spectrum is equispaced, with spacing ħ|Ω|; all the corresponding eigenfunctions are also exhibited. This finding obtains as special case of a more general (new) class of isochronous Hamiltonians.

Copyright

© 2010 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - F. Calogero
AU  - F. Leyvraz
PY  - 2021
DA  - 2021/01/07
TI  - Isochronous Oscillators
JO  - Journal of Nonlinear Mathematical Physics
SP  - 103
EP  - 110
VL  - 17
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925110000611
DO  - 10.1142/S1402925110000611
ID  - Calogero2021
ER  -