Hamiltonian periodic solution

J.H. Shirley, Solution of the Schrodinger equation with a Hamiltonian periodic in time, Phys. Rev. 138 (4B) (1965) B979. 

In ref. 165 a S5miplectic explicit RKN method for Hamiltonian systems with periodical solutions is obtained. The characteristics of this new method are ... 

Remark We shall prove later that in a Hamiltonian system there are always two unit eigenvalues, but only one eigenvector. So, there is only one periodic solution of the variational equations, corresponding to the double unit eigenvalue. 

We note now that the variational equations correspond to a periodic orbit x(t). So, f(t) = x(t) is a periodic solution of the variational equations and according to Section 4.1, one eigenvalue is equal to one, Ai = 1. Using now relation (40) we come to the conclusion that the monodromy matrix of a Hamiltonian system corresponding to a periodic orbit has a double unit eigenvalue. 

The exact solution of equations (64) is not easy to obtain analytically, but some insight can be gained by examining their equilibrium points (which correspond to periodic solutions of the two-degrees-of-freedom Hamiltonian < H >). They are defined by... 

In the supercell approach, the defect is instead enclosed in a sufficiently large unit cell and periodically repeated throughout space. A common problem with both approaches is the availability of high-level quantum-mechanical periodic solutions, because, as already mentioned, it is difficult to go beyond the one-electron Hamiltonian approximations (HF and DFT), at present. 

Proposition 2.1.2. Let a Hamiltonian system v on a nonsingular compact three-dimensional constant-energy surface Q be integrated by means of a Bott integral f, NoWf if the system v on Q has no stable periodic solutions then ... 

Let be a smooth symplectic manifold (compact or noncompact) and let V = sgradfT be a Hamiltonian system that is Liouville-integrable on a certain nonsingular compact three-dimensional constant-energy surface Q by means of a Bott integral /. Let m by the number of such periodic solutions of the system v on the surface Q on which the integral f attains a strictly local minimum or maximum (then the solutions are stable). Next, let p be the number of two-dimensional... 

Conley, C., and Zehnder, E. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math. 87 (1984), No. 2, 207-255. 

As a summary of this chapter it can be said that any periodic solution of the Hamiltonian of eq. (1) will result, independently of the size of Lc, in a hybridization gap. For an even count of f and d electrons (and no other electrons in the conduction band) the Fermi level will be in this gap and the materials will be intermediate-valent insulators at T 0. For an odd electron count the Fermi level will be in one of the density of states peaks above or below the gap, and the materials are intermediate-valent and heavy-fermion metals at 7 —> 0. In fig. 2 a simplified dispersion and density of states diagram is shown with most of the spectral weight in the f-like states. In fig. 2d the f-d hybridization extends over the whole Brillouin zone. The true experimental evidence of gaps in intermediate-valent insulators is about 12 years old (Batlogg et al. 1981), the experimental proof of gaps in heavy fermions is only 6 years old (Marabelli et al. 1986a). 

The general definitions may be applied with slight modifications./or appropriate generalization of the definitions/ in the multidimensional case. Hamiltonian and CRE systems can only be defined for an even number of variables. Some other notions can also be Introduced, the most Important among these is that div f=0, a property equivalent to being Hamiltonian in the two-dimensional case. This property and the existence of /global, time independent/ first integrals are closely connected with the existence of periodic solutions /Toth, 1987/ Several conjectures will be formulated here. 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

Molecules, in general, have some nontrivial symmetry which simplifies mathematical analysis of the vibrational spectrum. Even when this is not the case, the number of atoms is often sufficiently small that brute force numerical solution using a digital computer provides the information wanted. Of course, crystals have translational symmetry between unit cells, and other elements of symmetry within a unit cell. For such a periodic structure the Hamiltonian matrix has a recurrent pattern, so the problem of calculating its eigenvectors and eigenvalues can be reduced to one associated with a much smaller matrix (i.e. much smaller than 3N X 3N where N is the number of atoms in the crystal). 

Kaminski and Jorgensen (1998) have proposed one particularly simple QM/MM approach to address this problem, which tliey refer to as AMl/OPLS/CMl (AOC). In AOC, Monte Carlo calculations are canied out for solute molecules represented by the AMI Hamiltonian embedded in periodic boxes of solvent molecules represented by the OPLS force field. Thus, 7/qm in Eq. (13.1) is simply the AMI energy for the solute, and //mm is evaluated for all solvent-solvent interactions using the OPLS force field. The QM/MM interaction energy is computed in a fashion closely resembling the standard approach for MM non-bonded interactions... 

To anticipate the result of pulsed excitation of a superposition state, note from Eqs. (6.66) and (6.68) that the Hamiltonian is strictly periodic in time. We denote the time evolution operator associated with one period T as F. Although it is not possible to give an explicit form of F in the kicked molecule case, the existence j of this formal solution yields a stroboscopic description of the dynamics,... 

In solutions of water and surfactant, the surfactant monolayers can join, tail side against tail side, to form bilayers, which form lamellar liquid crystals whose bilayers are planar and are arrayed periodically in the direction normal to the bilayer surface. The bilayer thickens upon addition of oil, and the distance between bilayers can be changed by adding salts or other solutes. In the oil-free case, the hydrocarbon tails can be fluidlike (La) lamellar liquid crystal or can be solidlike (Lp) lamellar liquid crystal. There also occurs another phase, Pp, called the modulated or rippled phase, in which the bilayer thickness varies chaotically in place of the lamellae. Assuming lamellar liquid crystalline symmetry, Goldstein and Leibler [19] have constructed a Hamiltonian in which (1) the intrabilayer energy is calculated... 

We may think of a free-electron gas as having a vanishing potential (or equivalently, a constant potential, since wc can measure energies from that potential level). The Hamiltonian becomes simply -h V Ilm, and the solutions of the time-independent Schroedinger equation, Eq. (1-5), can be written as plane waves, e h Wc must apply suitable boundary conditions, and this is most conveniently done by imagining the crystal to be a rectangular parallelepiped, as shown in Fig. 15-1. Then wc apply periodic boundary conditions on the surface, as wc did following F.q. (2-2). The normalized plane-wave stales may be written as... 

The natural basis set for the above Hamiltonian is given by the 1 functions obtained from tlie direct product of the two basis states, DA) and D+A ) on each molecular site. The corresponding Hamiltonian matrix is easily diagonalized for clusters of finite dimension. Specifically, by exploiting translational symmetry, we obtained exact solutions for systems with up to 16 sites and periodic boundary conditions. 

In crystalline semiconductors, it is relatively easy to understand the formation of gaps in energy states of electrons and hence of the valence and conduction bands using band theory (see Ziman, 1972). Band structure arises as a consequence of the translational periodicity in the crystalline materials. For a typical crystalline material which is a periodic array of atoms in three dimensions, the crystal hamiltonian is represented by a periodic array of potential wells, v(r), and therefore is of the form, 7/crystai = ip l2m) + v(r), where the first term p l2rri) represents the kinetic energy. It imposes the eondition that the electron wave functions, which are solutions to the hamiltonian equation, H V i = E, Y, are of the form... 

The presence of the periodic potential f/(r) has important consequences with regard to the solutions of the time-independent Schrodinger equation associated with the Hamiltonian (4.71). In particular, a fundamental property of eigenfunctions of such a Hamiltonian is expressed by the Bloch theorem. 

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