Heun functions and their uses in physics (Q2837070)

The Heun equation is a second-order linear ODE with four Fuchsian singularities which, in its normalised form, can be written as NEWLINE\[NEWLINE \frac{d^2 w}{dz^2} + \left[ \frac{c}{z} + \frac{d}{z-1} + \frac{e}{z-f} \right] \frac{d w}{d z} - \frac{abz-q}{z(z-1)(z-f)} w = 0, NEWLINE\]NEWLINE where \(a+b+1=c+d+e\), the singularities being at \(0\), \(1\), \(f\) and \(\infty\). Confluent forms of this equation like the Mathieu or Lamé equations are obtained by the coalescence of two or more of the singularities, resulting in equations which also have irregular singularities.NEWLINENEWLINEThe article gives an extensive overview of examples of physical problems in the literature which can be reduced to Heun's equation or its confluent forms, many of these arise in quantum mechanics and general relativity. From his own work [\textit{T. Birkandan} and the author, J. Phys. A, Math. Theor. 40, No. 5, 1105--1116 (2007); corrigendum ibid. 40, No. 36, 11203 (2007; Zbl 1108.35102); J. Math. Phys. 48, No. 9, 092301, 11 p. (2007; Zbl 1152.81341)] the author then discusses the Dirac equation in the background of the Nutku helicoid metric, the solutions of which can be expressed, after separation of variables, by Mathieu functions. Another application by the author [\textit{T. Birkandan} and the author, J. Math. Phys. 49, No. 5, 054101, 10 p. (2008; Zbl 1152.81342)] are the scalar field equations in the background of the extended Eguchi-Hanson metric which can be solved in terms of the confluent Heun functions \(H_C\).NEWLINENEWLINEFor the entire collection see [Zbl 1255.00016].