Group Theory Approach to Band Structure: Scarf and Lamé Hamiltonians

Abstract

The group theoretical treatment of bound and scattering state problems is extended to include band structure. We show that one can realize Hamiltonians with periodic potentials as dynamical symmetries, where representation theory provides analytic solutions, or which can be treated with more general spectrum generating algebraic methods. We find dynamical symmetries for which we derive the transfer matrices and dispersion relations. Both compact and non-compact groups are found to play a role.

Temporally stable coherent states for infinite well and Pöschl–Teller potentials

“…Then, if λ 3/2 or κ 3/2, the periodized Hamiltonian H per R is self-adjoint, and has a pure point spectrum, with each eigenvalue of infinite multiplicity. On the contrary, if 1/2 < λ, κ 3/2, then H per R really looks as the Hamiltonian of a 1-D crystal, and indeed, it has no eigenvalue and its spectrum has a band structure, that is, it is purely continuous with infinitely many gaps [7,8,30].…”

“…The average position and average velocity of the particule are then 8) whereas the mean square dispersions are…”

“…Clearly this is a smooth approximation, for λ, κ → 1 + , of the infinite square-well over the interval [0, πa]. These potentials, called the Pöschl-Teller (PT) potentials [3,5], are shown in Figure 2 for the values (λ, κ) = (4, 4), (4,8), (4,16), respectively. In order to make contact with standard quantum mechanics on the whole line, there are two possibilities.…”

Critical Behavior of the Kramers Escape Rate in Asymmetric Classical Field Theories

“…Using the identity [22] (24) is the l = 3 Lamé equation [24,25] , a Schrödinger equation with periodic potential of period 2K(m). Its Bloch wave spectrum consists of four energy bands, and its eigenfunctions can be expressed in terms of Lamé polynomials [24] .…”

Bound states and band structure—A unified treatment through the quantum Hamilton–Jacobi approach

“…Quite sometime back, Scarf showed that a solvable model exists, which exhibits both discrete bound states and band spectra [9], as a function of the coupling parameter. The group theoretical aspects of this problem have recently been investigated [10]. The fact that Scarf potential yields both bound states and band structure, as a function of a coupling parameter, makes this model an ideal one to study the interplay of these two types of distinct behavior in a given quantal problem.…”