ASSOCIATED LAMÉ EQUATION, PERIODIC POTENTIALS AND sl(2, ℝ)

Abstract

We propose a new approach based on the algebraization of the Associated Lamé equationwithin sl(2,R) to derive the corresponding periodic potentials. The band edge eigenfunctions and energy spectra are explicitely obtained for integers m,ℓ. We also obtain the explicit expressions of the solutions for half-integer m and integer or half-integer ℓ.

Shape-invariant quantum Hamiltonian with position-dependent effective mass through second-order supersymmetry

“…The members of the other sequence { A † n ψ + 0,1 (x)} can be obtained by simply changing the sign of K 3 in the corresponding members of the sequence { A † n ψ + 0,2 (x)}. The nature of the function g(x) given by (35) is crucial for the normalizability of the wave functions. In the first place, m(x) must be so chosen that it verifies g 2 (x) → +∞ as x → ±∞ and remains finite otherwise.…”

Supersymmetric partners for the associated Lamé potentials

“…In some previous papers it has been studied the associated Lamé potentials and its SUSY partners for (m, ℓ) = (1, 1), (m, ℓ) = (2, 1), and (m, ℓ) = (3, 1) [3,4,13,14]. Here we will illustrate our general procedure for the associated Lamé potentials with (m, ℓ) = (3, 2), i.e.,…”

A new effective mass Hamiltonian and associated Lamé equation: bound states

“…In the intermediate region the reformulation of equation (2.18) is useful to solve the spectral problem. The crucial observation is that this equation may be transformed into the well-known associated Lamé equation [52][53][54][55][56][57] …”