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arXiv:cond-mat/9909294 · Physics Letters A

The energy spectrum of complex periodic potentials of the Kronig–Penney type

H.F Jones

Published Nov 1, 1999• 52 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(99)00672-6

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We consider a complex periodic PT-symmetric potential of the Kronig-Penney type, in order to elucidate the peculiar properties found by Bender et al. for potentials of the form V=i(sin⁡x)2N+1V=i(\sin x)^{2N+1}V=i(sinx)2N+1, and in particular the absence of anti-periodic solutions. In this model we show explicitly why these solutions disappear as soon as V∗(x)≠V(x)V^*(x)\neq V(x)V∗(x)=V(x), and spell out the consequences for the form of the dispersion relation.
Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.