PT-Invariant Periodic Potentials with a Finite Number of Band Gaps

Abstract

We obtain the band edge eigenstates and the mid-band states for the complex, PT-invariant generalized associated Lamé potentials $V^{PT}(x)=-a(a+1)m \sn^2(y,m)-b(b+1)m {\sn^2 (y+K(m),m)} -f(f+1)m {\sn^2 (y+K(m)+iK'(m),m)}-g(g+1)m {\sn^2 (y+iK'(m),m)}$, where $y\equiv ix+\beta$, and there are four parameters $a,b,f,g$. This work is a substantial generalization of previous work with the associated Lamé potentials $V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\sn^2 (x+K(m),m)}$ and their corresponding PT-invariant counterparts $V^{PT}(x)=-V(ix+\beta )$, 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 $V^{PT}(x)$ are periodic problems with a finite number of band gaps. Further, usingsupersymmetry, we construct several additional, new, 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.

30 pages, 0 figures

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Cited by in corpus (8)

• Making Sense of Non-Hermitian Hamiltonians
• Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
• Spectral singularities in PT-symmetric periodic finite-gap systems
• Conditional observability
• Generalized Swanson models and their solutions
• Complex Periodic Potentials with a Finite Number of Band Gaps
• The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations
• Quasi-Periodic Solutions of Heun's Equation