In two previous papers, the authors have introduced the concept of binormal differential equations. It has been shown that invariants of the Courant-Snyder type are associated to the scalar products of the column vector associated to an ordinary differential equation and to its binormal. In this paper, we show the equivalence of the above invariant and the Lewis form. We also introduce a density matrix for a second-order differential equation and clarify the geometrical meaning of the Twiss parameters. The importance of the above results in the analysis of quantum problems such as,e.g., the evolution of squeezed states is finally stressed.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
G. Dattoli, V. Loreto, C. Mari, M. Richetta andA. Torre:Nuovo Cimento B,106, 1357 (1991).
G. Dattoli, V. Loreto, C. Mari, M. Richetta andA. Torre:Nuovo Cimento B,106, 1375 (1991).
E. D. Courant andS. Snyder:Ann. Phys.,3, 1 (1958).
H. R. Lewis:J. Math. Phys.,9, 1976 (1968).
H. R. Lewis andW. B. Riesenfeld:J. Math. Phys.,10, 1458 (1969).
G. Dattoli, A. Dipace andA. Torre:Phys. Rev. A,33, 2789 (1986);P. K. Aravind:J. Opt. Soc. Am. B,5, 1545 (1988).
K. Wodkiewicz andJ. H. Eberly:J. Opt. Soc. Am. B,2, 158 (1985).
P. Caldirola:Nuovo Cimento,18, 393 (1941);E. Kanai:Prog. Theor. Phys.,3, 440 (1948).
H. Dekker:Phys. Rep.,80, 1 (1980).
G. Dattoli, S. Solimeno andA. Torre:Phys. Rev. A,35, 1668 (1987);G. Dattoli, P. Di Lazzaro andA. Torre:Nuovo Cimento B,105, 2, 165 (1990).
W. Hasse:J. Math. Phys.,16, 2005 (1975);J. Phys. A,11, 1245 (1978).
F. Sacherer: CERN Internal Report S11DL/70–12.
M. Sands:The Physics of Electron Storage Rings, an Introduction, SLAC-121 UC-28 (ACC).
K. Steffen:Basic Course on Accelerator Optics, CERN 85-19 (1985).
Rights and permissions
About this article
Cite this article
Dattoli, G., Loreto, V., Mari, C. et al. Biunitary transformations and ordinary differential equations.—III. Nuovo Cim B 106, 1391–1399 (1991). https://doi.org/10.1007/BF02728368
- Ordinary differential equations
- Infinite-dimensional Lie groups
- Propagation and transmission in homogeneous media
- Particle beams and particle optics