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Some exact results for mid-band and zero band-gap states of associated Lamé potentials
1.For the properties of Jacobi elliptic functions, see, for example, I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980). The modulus parameter is often called in the mathematics literature. The related complementary quantity is often called
2.A. Khare and U. Sukhatme, J. Math. Phys. 40, 5473 (1999); hereafter, we will refer to this paper as I.
3.W. Magnus and S. Winkler, Hill’s Equation (Wiley, New York, 1966).
4.F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, 1981); E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, 1980).
5.See, for example, W. Magnus and S. Winkler, Hill’s Equation (Wiley, New York, 1966), Chap. 7.
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