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Such a reinterpretation was already sought by Dyson but could be implemented only recently on the basis of a discovery of Pechukas [32]. The fate of the N eigenvalues and eigenvectors of an N × N Hermitian matrix, H = H0 + λV, is in one-to-one correspondence with the classical Hamiltonian dynamics of a particular one-dimensional N -particle system upon changing the weight λ of a perturbation V [32–34]. This fictitious system, now often called Pechukas–Yukawa gas, has λ as a time, the eigenvalues E n of H as coordinates, the diagonal elements Vnn of the perturbation V in the H representation as momenta, and the off-diagonal elements Vnm related to certain angular momenta.

2) The motivation for this definition is that many Hamiltonians of practical importance are invariant under conventional time reversal, [H, T ] = 0. An atom and a molecule in an isotropic environment, for instance, have Hamiltonians of that symmetry. But, as already mentioned in Sect. 1, conventional time reversal is broken by an external magnetic field. In identifying the canonical transformations of Hamiltonians from their symmetries in Sects. 1). 1) is not at all necessary for the above classification of Hamiltonians according to their group of canonical transformations.

The Floquet operator, being unitary, has unimodular eigenvalues (involving eigenphases alias quasi-energies) and mutually orthogonal eigenvectors, FΦν = e−iφν Φν , Φμ |Φν = δμν . 8) I shall in fact be concerned only with normalizable eigenvectors. With the eigenvalue problem solved, the stroboscopic dynamics can be written out explicitly, e−inφν Φν |ψ(0) Φν . 9) ν Monochromatic perturbations are relatively easy to realize experimentally. Much easier to analyse are perturbations for which the temporal modulation takes the form of a periodic train of delta kicks, +∞ H (t) = H0 + λV δ(t − nτ ).

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