Floquet theorem

According to the Floquet theorem [Arnold 1978], this equation has a pair of linearly-independent solutions of the form x(z,t) = u(z, t)e p( 2nizt/p), where the function u is -periodic. The solution becomes periodic at integer z = +n, so that the eigenvalues e we need are = ( + n). To find the infinite product of the we employ the analytical properties of the function e z). It has two simple zeros in the complex plane such that... 

S.-I. Chu, D.A. Telnov, Beyond the Floquet theorem Generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields, Phys. Rep. 390 (2004) 1. 

Unlike the above mentioned methods, another Floquet-theorem-based approach, the many-electron many-photon theory (MEMPT) of Mercouris and Nicolaides (71,72) does not involve complex rotated Hamiltonians. The complex coordinate rotation is used only to regularize that part of the wave functions which describes unbound electrons (see the CESE method). This allows efficient description of bound or quasi-bound states, involved in a problem under consideration, by MCHF solutions and therefore enables ab initio application to many-electron systems (71,72,83-87). 

If the linear equation (11.16) has periodically varying coefficients with period T, A(t + T) = A(t), the Floquet theorem provides the fundamental result that the fundamental matrix of (11.16) can be written as the product of a T-periodic matrix and a (generally) nonperiodic matrix [458, p. 80]. 

The universal structural feature of perfect crystals is their periodicity. An immediate consequence of this periodicity is the Bloch-Floquet theorem stating that the one-electron eigenfunctions are of the form... 

The title of this subsection refers to a recent paper by Kylstra and Joachain investigating double poles of the S matrix [25]. Their paper is based on the time-dependent Lippmann-Schwinger equation. Since the Hamiltonian is periodic in time the use of the Floquet theorem permits one to apply the time-independent theory of scattering. As in Ref. [25] we consider two quasi bound states (n = 2). Instead of starting from an Hamiltonian periodic in time (semi-classical approximation) we use a time-independent model. The laser field is assumed to be quantized and as a result the total Hamiltonian describing the atom in the laser field is time-independent (see chapter VI of Ref. [13]) Our aim is twofold To reproduce the results of Ref. [25] and, more generally, to illustrate the relevance of simple models to describe collision... 

We first consider an analysis of physical systems in periodic external fields using the Floquet theorem [134, 168]. As we shall see below, the theorem provides theoretical basis for the existence of field-dressed quasi-stationary state which expand the propagator. Earlier application of the Floquet theorem or related ideas to physical problems includes Refs. [224, 370], whereas the progress in this field is recently reviewed in Ref. [90]. Although it will later be extended to allow small non-periodic modulations, discussions in this section assumes perfect periodicity. The Schrodinger equation is given as... 

Proof of the Floquet theorem Readers not familiar with the Floquet theorem are recommended to refer to textbooks on ordinary differential equations such as Ref. [168], where one would find a formal proof of the theorem different from the following discussions. Here we try to derive an intuitive but physically appealing explanation of the Floquet theorem based on Fourier analysis. We start with the Fourier transformation of the time dependent Schrodinger equation (8.1), ... 

The Floquet theorem, when apphed to the quantum mechanics [370], implies the stationarity of Floquet states imder a perfectly periodic Hamiltonian. We define the electronic Floquet operator as 7ft = Hf — ihdt and the Floquet states as its periodic eigenstates which satisfy Tlt x t)) = A] A(f))- The above mentioned stationarity states that the solution of time-dependent Schrodinger equation ] t) can be expanded as... 

Unfortunately, however, the original form of Floquet theorem does not apply to the cases where H is not perfectly periodic. This includes the dynamics under pulsed laser field and/or d3mamics coupled with nuclear motion, which are the cases we are most interested in. In order to apply the Floquet based analyses to such cases, one has to generalize the original formulation. Interestingly, as we will show below, the effects of nonperiodicities can be incorporated in terms of nonadiabatic transitions among Floquet states. 

This condition is sufficient to apply the Floquet theorem (or Floquet-Liapounoff theorem), " which states that U satisfies the form of equations (42)-(45) ... 

With a specific design of the traditional FSS, as a result of the Floquet theorem, it is possible to select a unique frequency value for the filter. By solving Maxwell equations, it is possible to determine the theoretical electromagnetic FSS behavior. However, considering the complexity of the phenomena involved, it is only possible to perform a realistic characterization of the FSS with an experimental test. 

The tubular current source was described in Section 21-6, where we showed that it is ineffective in exciting bound modes unless either of the resonance conditions of Eq. (21-15) is satisfied. A similar conclusion holds for the radiation fields. If the tube length 2L is large compared to the spatial period 2n/Sl, where 2 is the frequency in Eq. (21-13), it is intuitive that power will be radiated essentially at a fixed angle to the fiber axis. This is also a consequence of Floquets theorem [7]. However, unlike the current dipole, radiation now depends on the orientation of the currents on the tube. 

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