Perturbation strength

The derivative formulation is perhaps the easiest to understand. In this case the energy is expanded in a Taylor series in the perturbation strength A. 

Derivative techniques consider the energy in the presence of the perturbation, perform an analytical differentiation of the energy n times to derive a formula for the nth-order property, and let the perturbation strength go to zero. 

For real wave functions the first and third terms are identical. Letting the perturbation strength go to zero yields... 

In the limit of the perturbation strength going to zero this reduces to... 

The index 0 indicates that the derivatives are taken for zero perturbation strengths. The equations for the first-order amplitudes and multipliers are obtained as ... 

This equation describes the electronic reaction to the oscillating external perturbation. In principle, it has a solution for any frequency co. One special class of solutions is, however, of particular interest. If at a frequency co, there is a solution i//(l that satisfies the above equation also at zero perturbation strength (hp — 0), then the unperturbed system will also be stable in the state described by this particular solution + hf/ K In such circumstances the term X is no longer bound to the perturbation strength. Instead, it can take any value, as long as it is sufficiently small to remain in the linear response region of the system. Such a new state, however, is nothing but an excited state. 

The F matrix and -q vector in equation (4) are defined as the second partial derivatives of the Lagrangian, taken at the zero perturbation strength ... 

Second-order molecular properties can be defined as second derivatives of the (time-averaged) quasienergy Q with respect to frequency-dependent perturbation strengths b (wb) at zero perturbation (s=0)... 

Figure 4.19 Transfer infidelity 1 - F T) for a modulated boundary-controlled coupling a tt) = a sin C y) as a function of (a) the transfer time T. (inset) the maximum value of the boundary coupling and (b) the perturbation strength Cj of the noisy channel, averaged over 10 noise realizations for = 0.6 and = 0.7. In static noisy channels, the infidelity obtained under static control p = 0 (empty circles) is shown to be strongly reduced under dynamical p = 2 control (empty squares). A fluctuating noisy channel is less damaging in the Markovian Unfit where the correlation time of the noise fluctuations 0 (p = 0, green solid circles), the infidelity converges to its unperturbed value.

The KAM theorem demonstrates the existence of KAM tori when the perturbations to the motion are small. What happens when a nearly integrable Hamiltonian is strongly perturbed For example, with increasing perturbation strength, what is the last KAM torus to be destroyed and how should we characterize the phase space structures when all KAM tori are destroyed Using simple dynamical mapping systems, which can be regarded as Poincare maps in Hamiltonian systems with two DOFs, MacKay, Meiss, and Percival [8,9] and Bensimon and Kadanoff [10] showed that the most robust KAM curve... 

Let s consider the effect of perturbation on classical trajectories in the complex domain. Since the perturbation changes periodically in the real-time domain, namely a sinusoidal function, the effective perturbation strength in the complextime domain is amplified exponentially that is, e —> ee l l (see Fig. 3b). 

Figure 3. Effects of the periodic perturbation, (a) Integration path on the complex time plane. (b) Deformation of the potential by the periodical perturbation. In the case where Im t = Im ti 0— that is, the part of integration path indicated by the same broken line in (a)— the oscillation of complexified potential is amplified exponentially as shown by the broken lines, (c) Change of the tunneling trajectory with increase of the perturbation strength. In the bottom figure, a trajectory stating at ti in the close neighborhood of t c is drawn.

Using the dehnition of the parameter p given by Eq. (B.5), the intersection t c is decided by the above relations, (B.8) and (B.9). When E is considerably less than 1, the real intersections (i.e.,Imtic = Imp = 0) exist only for a sufficiently strong perturbation strength such that e > dih = 1(1 Ei)/ -x( ))l>butifthe intersection is allowed to be complex, it may exist at an arbitrarily weak perturbation strength. Indeed, the complex intersections... 

In the case of the kicked rotor we were able to predict the critical perturbation strength by applying the Chirikov overlap criterion. This criterion does two things for us. First, it provides us with an excellent physical picture which explains qualitatively the mechanism of the chaos transition secondly, it provides us with an analytical estimate for the critical field. 

A General perturbation strength At Small tfinitet timestep... 

Lagrange multiplier General perturbation strength Hessian shift parameter Angular momentum quantum number Lagrange function... 

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