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By using and browsing our websites, you consent to cookies being used in accordance with our policy. If you do not consent, you must disable cookies or refrain from using the site. The length of the string is L. Hook and Karta Singh Kooner The Hamiltonian H for this system is a constant of the motion and thus the energy E is a time-independent quantity.
This is the periodic analog of Figures 2. Five trajectories are shown for each cell. The trajectories shown here are the periodic analogs of the trajectories shown in Figures 2. If the energy of the classical particle in a periodic potential is taken to be complex, the particle begins to hop from well to well in analogy to the behavior of the particle in Figure 2. This hopping behavior is displayed in Figure 2. A tunneling trajectory for the Hamiltonian 2. The classical particle hops from well to well in a random-walk fashion. The particle starts at the origin and then hops left, right, left, left, right, left, left, right, right.
This is the sort of behavior normally associated with a particle in a crystal at an energy that is not in a conduction band. At the end of this simulation the particle is situated to the left of its initial position. The trajectory never crosses itself. The most interesting analogy between quantum mechanics and complex classical mechanics is established by showing that there exist narrow conduction bands in the periodic potential for which the quantum particle 11 Complex elliptic pendulum exhibits resonant tunneling and the complex classical particle exhibits unidirectional hopping [35, 38].
This qualitative behavior is illustrated in Figure 2. A classical particle exhibiting a behavior analogous to that of a quantum particle in a conduction band that is undergoing resonant tunneling. Unlike the particle in Figure 2. These Figure 2. Complex-energy plane showing those energies that lead to tunneling hopping behavior and those energies that give rise to conduction.
Hopping behavior is indicated by a hyphen - and conduction is indicated by an X. In most places the resolution distance between points is 0. The regions indicated by arrows are blown up in Figure 2. Detailed portions of the complex-energy plane shown in Figure 2. Note that the edge of the conduction band, where tunneling hopping behavior changes over to conducting behavior, is very sharp. Elliptic potentials are natural doubly-periodic generalizations of trigonometric potentials. When 0 Figure 3. In Figure 3.
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This trajectory is superimposed on a plot of the real part of the cosine potential. The particle oscillates horizontally. The ensuing wavy vertical motion passes close to many poles before the particle gets captured by another pair of turning points. The particle winds inwards and outwards around these turning points and eventually returns to the original pair of turning points. After escaping from these turning points again, the particle now moves in the positive-imaginary direction.
Figure 3. In the background is a plot of the real part of the cosine potential, which is shown in detail in Figure 3. It is clear that classical trajectories associated with doubly periodic potentials have an immensely interesting structure and should be investigated in much greater detail to determine if there is a behavior analogous to band structure shown in Figures 2. The trajectory spirals outward around the two turning points and, when it gets very close to a pole, it suddenly begins to travel downward in a wavy fashion in the negative-imaginary direction.
It eventually gets very close to a simple pole, gets trapped, and spirals inward towards a pair of turning points.
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After spiraling inwards, it then spirals outwards never crossing itself and goes upward along a path extremely close to the downward wavy path. It is then recaptured by the original pair of turning points in the central pane. Evidently, the particle trajectory strongly prefers to move vertically upward and downward, and not horizontally. The vertical motion distinguishes cnoidal trajectories from those in Figures 2.
Quantum mechanics is essentially wavelike; probability amplitudes are described by a wave equation and physical observations involve such wavelike phenomena as interference patterns and nodes. In contrast, classical mechanics describes the motion of particles and exhibits none of these wavelike features. Nevertheless, there is a deep connection between quantum mechanics and complex classical mechanics.
In the complex domain the classical trajectories exhibit a remarkable behavior that is analogous to quantum tunneling. Periodic potentials exhibit a surprising and intricate feature that closely resembles quantum band structure. Hook and Karta Singh Kooner Our early work on singly periodic potentials strongly suggests that further detailed analysis should be done on doubly periodic potentials.
Doubly periodic potentials are particularly interesting because they have singularities.
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Two important and so far unanswered questions are as follows: i Does a complex classical particle in a doubly periodic potential undergo a random walk in two dimensions and eventually visit all lattice sites? The CPT norm is strictly real and positive and thus the theory is associated with a conventional Hilbert space.
The time evolution is unitary because it preserves the CPT norms of vectors. A detailed discussion of these features of PT quantum mechanics is presented in . The eigenvalues are real and positive see . There is extremely strong numerical evidence that the set of eigenfunctions form a complete basis, but to our knowledge this result has not yet been rigorously established.
The term unbounded below cannot be used in this context because the complex numbers are not ordered. PT quantum mechanics has many qualitative features, such as arbitrarily fast time evolution, that distinguish it from conventional Dirac-Hermitian quantum mechanics.
These features are discussed in . A: Math. YANG, Ann. K IP, Nature Phys. W U, Phys. A 81, DARG , and K. DARG, J. Hook and Karta Singh Kooner  C.
Costin, O. (Ovidiu) 1960-
H OOK, J. WANG, Phys. Unstable states are characteristic of open quantum systems, while in this paper the quantum systems discussed are closed systems. Brizard, Eur. Parabolic attitude Filippo Bracci Abstract. Being parabolic in complex dynamics is not a state of fact, but it is more an attitude.
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In these notes we explain the philosophy under this assertion. Although the germ might be non-invertible, here we will concentrate only on holomorphic diffeomorphisms. Let F denote such a germ of holomorphic diffeomorphism in a neighborhood of the origin 0 in Cn. In this case the map is topologically conjugate to its differential by the Hartman-Grobman theorem [21,22,27] and the dynamics is then completely clear.
However, already in case when all eigenvalues have modulus different from 1, holomorphic linearization is not always possible due to the presence of resonances among the eigenvalues see, for instance, [7, Chapter IV]. The very rough idea for germs tangent to identity is to consider the changes of Fatou coordinates on the intersection between an attracting petal and the subsequent repelling petal. These invariants, together with the multiplicity and the index, are the sought complete system of holomorphic invariants.
In such a case the germ is always formally linearizable, but, as strange as it might be, it is holomorphically linearizable if and only if topologically linearizable and this last condition is related to boundedness of the orbits in a neighborhood of 0. Non-linearizable elliptic germs present very 21 Parabolic attitude interesting dynamics that we are not going to describe in details here, leaving it to the reader to check the survey papers [2, 8, 9]. In higher dimension, the situation is much more complicated.
There is not such a clear distinction between parabolic or elliptic germs. We will concentrate on the parabolic behavior. This is the prototype of parabolic dynamics.
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