Exact solutions of the 1D Schrödinger equation with the Mathieu potential

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Abstract

© 2020 Elsevier B.V. The exact solutions of the 1D Schrödinger equation with the Mathieu potential V(x)=a2sin2⁡(bx)−ab(2c+1)cos⁡(bx)+vb2 (a>0,b>0) are presented as a confluent Heun function HC(α,β,γ,δ,η;z). The eigenvalues are calculated precisely by solving the Wronskian determinant. The wave functions for the positive and negative parameter c, which correspond to two different potential wells with symmetric axis x=0 and x=π are plotted. It is found that the wave functions are shrunk to the origin for given values of the parameters a=1,b=1 and v=2 when the potential parameter |c| increases.

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