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Exact solutions of the 1D Schrödinger equation with the Mathieu potential. (English) Zbl 1448.81313

Summary: The exact solutions of the 1D Schrödinger equation with the Mathieu potential \(V(x)=a^2\sin^2(bx)-ab(2c+1)\cos(bx)+vb^2(a>0,b>0)\) are presented as a confluent Heun function \(H_C(\alpha,\beta,\gamma,\delta,\eta;z)\). 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 \(x=\pi\) 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.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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