# §31.17(i) Addition of Three Quantum Spins

The problem of adding three quantum spins $\mathbf{s}$, $\mathbf{t}$, and $\mathbf{u}$ can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. We use vector notation $[\mathbf{s},\mathbf{t},\mathbf{u}]$ (respective scalar $(s,t,u)$) for any one of the three spin operators (respective spin values).

Consider the following spectral problem on the sphere $S_{2}$: $\mathbf{x}^{2}=x_{s}^{2}+x_{t}^{2}+x_{u}^{2}=R^{2}$.

 31.17.1 $\displaystyle\mathbf{J}^{2}\Psi(\mathbf{x})$ $\displaystyle\equiv(\mathbf{s}+\mathbf{t}+\mathbf{u})^{2}\Psi(\mathbf{x})=j(j+% 1)\Psi(\mathbf{x}),$ $\displaystyle H_{s}\Psi(\mathbf{x})$ $\displaystyle\equiv(-2\mathbf{s}\cdot\mathbf{t}-(\ifrac{2}{a})\mathbf{s}\cdot% \mathbf{u})\Psi(\mathbf{x})=h_{s}\Psi(\mathbf{x}),$

for the common eigenfunction $\Psi(\mathbf{x})=\Psi(x_{s},x_{t},x_{u})$, where $a$ is the coupling parameter of interacting spins. Introduce elliptic coordinates $z_{1}$ and $z_{2}$ on $S_{2}$. Then

 31.17.2 $\frac{x_{s}^{2}}{z_{k}}+\frac{x_{t}^{2}}{z_{k}-1}+\frac{x_{u}^{2}}{z_{k}-a}=0,$ $k=1,2$,

with

 31.17.3 $\displaystyle x_{s}^{2}$ $\displaystyle=R^{2}\frac{z_{1}z_{2}}{a},$ $\displaystyle x_{t}^{2}$ $\displaystyle=R^{2}\frac{(z_{1}-1)(z_{2}-1)}{1-a},$ $\displaystyle x_{u}^{2}$ $\displaystyle=R^{2}\frac{(z_{1}-a)(z_{2}-a)}{a(a-1)}.$ Symbols: $z$: complex variable, $a$: complex parameter, $R^{2}$: sphere and $x_{s}$, $x_{t}$, $x_{u}$: coordinates Permalink: http://dlmf.nist.gov/31.17.E3 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

The operators $\mathbf{J}^{2}$ and $H_{s}$ admit separation of variables in $z_{1},z_{2}$, leading to the following factorization of the eigenfunction $\Psi(\mathbf{x})$:

 31.17.4 $\Psi(\mathbf{x})=(z_{1}z_{2})^{-s-\frac{1}{4}}((z_{1}-1)(z_{2}-1))^{-t-\frac{1% }{4}}\*((z_{1}-a)(z_{2}-a))^{-u-\frac{1}{4}}w(z_{1})w(z_{2}),$

where $w(z)$ satisfies Heun’s equation (31.2.1) with $a$ as in (31.17.1) and the other parameters given by

 31.17.5 $\displaystyle\alpha$ $\displaystyle=-s-t-u-j-1,$ $\displaystyle\beta$ $\displaystyle=j-s-t-u,$ $\displaystyle\gamma$ $\displaystyle=-2s,$ $\displaystyle\delta$ $\displaystyle=-2t,$ $\displaystyle\epsilon$ $\displaystyle=-2u;$ $\displaystyle q$ $\displaystyle=ah_{s}+2s(at+u).$

For more details about the method of separation of variables and relation to special functions see Olevskiĭ (1950), Kalnins et al. (1976), Miller (1977), and Kalnins (1986).

# §31.17(ii) Other Applications

Heun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)).

For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. More applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). For application of biconfluent Heun functions in a model of an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere see Boyd and Natarov (1998).