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Complex periodic potentials with a finite number of band gaps. (English) Zbl 1112.81031
Summary: We obtain several new results for the complex generalized associated Lamé 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 \(y\equiv x-K(m)/2-iK'(m)/2\), \(sn(y,m)\) is the 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 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 and \(a\geq b\geq f\geq g\geq 0\)? 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.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
33E05 Elliptic functions and integrals
34A05 Explicit solutions, first integrals of ordinary differential equations
81U15 Exactly and quasi-solvable systems arising in quantum theory
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