General Periodic Potential Research Articles

Solution of a one-dimensional torsion Schrödinger equation with a general periodic potential

Relations are found for the calculation of eigenvalues and eigenfunctions of the Hamiltonian of internal rotational motion in molecules in the basis of plane waves. The dependences of kinematic coefficient F(φ) and potential V(φ) on dihedral angle φ are represented by Fourier series for both symmetric and asymmetric functions, as well as for general periodic functions. If a molecule has symmetry elements, the found solution transforms to that previously known.

Enhanced quantization on the circle

We apply the quantization scheme introduced in Klauder 2012 (arXiv:1204.2870) to a particle on a circle. We find that the quantum action functional restricted to appropriate coherent states can be expressed as the classical action plus ℏ-corrections. We succeed in proving this for a Hamiltonian endowed with a general periodic potential. This result extends the examples presented in the cited paper for a system on a nontrivial configuration space topology.

Gap solitons and their linear stability in one-dimensional periodic media

An analytical theory utilizing exponential asymptotics is presented for one-dimensional gap solitons that bifurcate from edges of Bloch bands in the presence of a general periodic potential. It is shown that two soliton families bifurcate out from every Bloch-band edge under self-focusing or self-defocusing nonlinearity, and an asymptotic expression for the eigenvalues associated with the linear stability of these solitons is derived. The locations of these solitons relative to the underlying potential are determined from a certain recurrence relation, that contains information beyond all orders of the usual perturbation expansion in powers of the soliton amplitude. Moreover, this same recurrence relation decides which of the two soliton families is unstable. The analytical predictions for the stability eigenvalues are in excellent agreement with numerical results.

QUANTUM-RESONANCE RATCHETS: THEORY AND EXPERIMENT

A theory of quantum ratchets for a particle periodically kicked by a general periodic potential under quantum-resonance conditions is developed for arbitrary values of the conserved quasimomentum β. A special case of this theory is experimentally realized using a Bose–Einstein condensate (BEC) exposed to a pulsed standing light wave. While this case corresponds to completely symmetric potential and initial wave-packet, a purely quantum ratchet effect still arises from the generic noncoincidence of the symmetry centers of these two entities. The experimental results agree well with the theory after taking properly into account the finite quasimomentum width of the BEC. This width causes a suppression of the ratchet acceleration occurring for "resonant" β, so that the mean momentum saturates to a finite ratchet velocity, strongly pronounced relative to that for nonresonant β.

Reply to "Comment on 'Energy-band equation for a general periodic potential' "

It is shown, by means of simple, soluble examples, that the argument of van Ek and Lodder’s preceding Comment is erroneous, as are their conclusions.Received 27 January 1989DOI:https://doi.org/10.1103/PhysRevB.40.1981©1989 American Physical Society

Comment on "Energy-band equation for a general periodic potential"

The new energy-band equation by Badralexe and Freeman is shown to be an identity.Received 18 July 1988DOI:https://doi.org/10.1103/PhysRevB.40.1979©1989 American Physical Society

Reply to "Comment on 'Energy-band equation for a general periodic potential' "

It is shown that the central proof of the Comment [Robert G. Brown and Mikael Ciftan, this issue, Phys. Rev. B 39, 10 415 (1989)] relies on a nonconvergent integral and hence is without foundation.Received 22 December 1988DOI:https://doi.org/10.1103/PhysRevB.39.10420©1989 American Physical Society

Semiclassical treatment of the periodic potential

The use of a general periodic potential is addressed using both semiclassical path integral and instanton techniques. Identical band structure is shown to result from either approach.

Eigenvalue equation for a general periodic potential and its multipole expansion solution.

It is shown that the Bloch function, as an element of the Hilbert space spanned by Bloch-periodic planes waves, can be also represented as an on-shell superposition of Bloch-periodic orbitals which, however, exist in a distribution sense. By using this result and the multipole expansion of the Bloch function at the origin, it is further shown that the integral eigenvalue equation of the Bloch function for a general periodic potential is equivalent (both necessary and sufficient) to an algebraic system of homogeneous linear equations for the coefficients of the multipole expansion of the Bloch function at the origin, akin to the much simpler case of a finite-range potential. In contrast to the Korringa-Kohn-Rostoker (KKR) equation, the contribution of the cell potential (whether of the muffin-tin or general form) introduces a supplementary structure dependence. However, the separation between structure and potential, typical for the KKR equation, can be restored by introducing various approximations.

Friedel-type sum rule for a general periodic potential

By using the continual direct-sum decomposition obtained previously for the t operators, it is shown that the change of the density of states in the presence of a general (non-muffin-tin) periodic potential can be given in a determinant form involving only the on-shell t matrix. The result obtained is close to the usual Friedel sum rule of scattering theory which, however, relies strongly on the existence of asymptotic free motion and hence, on the existence of the phase-shift motion. Thus, since the periodic potential does not allow for the existence of asymptotic free motion, the present result illustrates a rather general trace property of the resolvent, which appears to be independent of the very different boundary conditions in the two cases.

Energy-band equation for a general periodic potential.

A new approach---one not based on multiple scattering or the variational principle---to the problem of the band structure for a general (i.e., non-muffin-tin) periodic potential is presented. From the eigenvalue equation, the Bloch function is shown to be expandable as a multipole series around the origin and the coefficients are found to be given as functionals of the Bloch function and the cell potential. By introducing into this functional various representations of the Bloch function (as a superposition in which each individual term is not Bloch periodic), we obtain previously derived band-structure equations (some claimed to be exact). By deriving, however, a new representation for the Bloch function (as an on-shell superposition in which each term is Bloch periodic) and using this representation in the above functional, we obtain a new energy-band equation in which the potential can no longer be separated from the structure. When approximations are introduced, then Korringa-Kohn-Rostoker-type equations (in which the potential is separated from the structure) may be obtained. All the steps of this new approach are illustrated in the soluble case of a constant periodic potential. It is shown, in this case, that only the newly introduced band-structure equation is able to generate the correct eigenvalues.

T operators and their matrix elements for a general periodic potential.

A new approach for the treatment of the T operator is provided which cures the difficulties of the multiple-scattering approach to the general (non-muffin-tin) periodic potential problem. Solely from potential periodicity, i.e., without discriminating between muffin-tin and non-muffin-tin cases, the T operators are shown to admit a direct integral decomposition in terms of the reduced T operators. This feature is further exploited by introducing ``Bloch periodic scattering states'' which finally results in a closed (compact) expression for the on-shell matrix elements. In the L representation, their functional form is related to that derived within multiple-scattering theory for the muffin-tin case but irrespective of whether the potential is muffin tin or not, the structure dependence cannot be separated from that of the potential (as known in the multiple-scattering approach for the muffin-tin case). However, this separation can be restored by introducing various approximations which partially break the Bloch periodicity. Hence, the separation between structure and potential, even in the muffin-tin case, is shown to be an approximate result.