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PT-invariant periodic potentials with a finite number of band gaps. (English) Zbl 1110.81158
Summary: We obtain the band edge eigenstates and the midband states for the complex, generalized associated Lamé potentials $$V^{\text{PT}}(x)=-a(a+1)m\text{sn}^2 (y,m)- b(b+1)m\text{sn}^2 (y+K(m),m)-f (f+1)m\text{sn}^2 (y+K(m)+iK'(m),m)- g(g+1)m\text{sn}^2 (y+iK'\times(m),m)$$, where $$y\equiv ix+\beta$$, and there are four parameters $$a$$, $$b$$, $$f$$, $$g$$. By construction, this potential is PT-invariant since it is unchanged by the combined parity (P) and time reversal (T) transformations. This work is a substantial generalization of previous work with the associated Lamé potentials $$V(x)= a(a+1)m \operatorname{sn}^2(x,m)+b(b+1)m\text{sn}^2(x+K(m),m$$) and their corresponding PT-invariant counterparts $$V^{\text{PT}}(x)= -V(ix+\beta)$$, both of which involving just two parameters $$a,b$$. We show that for many integer values of $$a,b,f,g$$, the PT-invariant potentials $$V^{\text{PT}}(x)$$ are periodic problems with a finite number of band gaps. Further, using supersymmetry, we construct several additional, complex, PT-invariant, periodic potentials with a finite number of band gaps. We also point out the intimate connection between the above generalized associated Lamé potential problem and Heun’s differential equation.

##### MSC:
 81U15 Exactly and quasi-solvable systems arising in quantum theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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