Complex Periodic Potentials with a Finite Number of Band Gaps

Abstract

We obtain several new results for the complex generalized associated Lame potential V(x)= a(a+1)m sn^2(y,m)+ b(b+1)m sn^2(y+K(m),m) + f(f+1)m sn^2(y+K(m)+iK'(m),m)+ g(g+1)m sn^2(y+iK'(m),m), where y = x-K(m)/2-iK'(m)/2, sn(y,m) is a Jacobi elliptic function with modulus parameter m, and there are four real parameters a,b,f,g. First, we derive two new duality relations which, when coupled with a previously obtained duality relation, permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the four parameters a,b,f,g. Second, we pose and answer the question: how many independent potentials are there with a finite number "a" of band gaps when a,b,f,g are integers? For these potentials, we clarify the nature of the band edge eigenfunctions. We also obtain several analytic results when at least one of the four parameters is a half-integer. As a by-product, we also obtain new solutions of Heun's differential equation.

arXiv comment33 pages, 0 figures

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Cited by in corpus (9)

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• Spectral singularities in PT-symmetric periodic finite-gap systems
• Talbot self-imaging in $\mathcal{P}\mathcal{T}$-symmetric complex crystals
• Middle Convolution and Heun's Equation
• Quasi-exact solvability in a general polynomial setting
• Integral transformation of Heun's equation and some applications
• Superoscillatory PT-symmetric potentials
• A duality of the classical action yields a reflection symmetry of the quantum energy spectrum