Analytically Solvable PT-Invariant Periodic Potentials

Abstract

Associated Lamé potentials $V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\cn^2 (x,m)}/{\dn^2(x,m)}$ are used to construct complex, PT-invariant, periodic potentials using the anti-isospectral transformation $x\to ix+\beta$, where $\beta$ is any nonzero real number. These PT-invariant potentials are defined by $V^{PT}(x)\equiv -V(ix+\beta )$, and have a different real period from $V(x)$. They are analytically solvable potentials with a finite number of band gaps, when $a$ and $b$ are integers. Explicit expressions for the band edges of some of these potentials are given. For the special case of the complex potential $V^{PT}(x)=-2m\sn^2(ix+β,m)$, we also analytically obtain the dispersion relation. Additional new, solvable, complex, PT-invariant, periodic potentials are obtained by applying the techniques of supersymmetric quantum mechanics.

12 pages, 3 figures

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Cited by in corpus (14)

• Making Sense of Non-Hermitian Hamiltonians
• Biorthogonal Quantum Systems
• Spectral singularities in PT-symmetric periodic finite-gap systems
• Periodic Potentials and Supersymmetry
• Generalized Swanson models and their solutions
• Self-isospectral tri-supersymmetry in PT-symmetric quantum systems with pure imaginary periodicity
• Supersymmetric solution of PT-/non-PT-symmetric and non-Hermitian Morse potential is studied to get real and Supersymmetric Solution of PT-/Non-PT-Symmetric and Non-Hermitian Morse Potential via Hamiltonian Hierarchy Method
• Absorption in atomic wires
• Complex Periodic Potentials with a Finite Number of Band Gaps
• PT-Invariant Periodic Potentials with a Finite Number of Band Gaps
• Supersymmetric Biorthogonal Quantum Systems
• Hierarchy of QM SUSYs on a Bounded Domain
• Superoscillatory PT-symmetric potentials
• Stability boundaries of a Mathieu equation having PT symmetry