Energy eigenvalues ofNexcitons and one hole on a Bethe lattice

A continued fraction is used to express the energy eigenvalues of the ($2N+1$)-body problem consisting of $N$ immobile and strongly bound Frenkel excitons plus one spinless hole (or electron) on a Bethe lattice of arbitrary coordination $K>~2$. For the "trion" ($N=1\mathrm{}$), the critical bandwidth ratio ${\nu }_c^T\equiv {\left(\frac{V}{C}\right)}_{\mathrm{crit}}$ for a bound state to exist is given by ${\nu }_c^T=\frac{1}{\sqrt{K-1\mathrm{}}}$, with $V$ and $C$ the valence- and conduction-band hopping matrix elements, respectively. For $N>\mathrm{}1\mathrm{}$, the critical ${\nu }_c$ depends on the geometry and number of excitons if $K>\mathrm{}2\mathrm{}$; in one dimension ($K=2\mathrm{}$), ${\nu }_c$ is always one. Because of the absence of closed loops on the Bethe lattice, the effective mass of the ($2N+1$)-body complex is always infinite. Residual longer-range interactions between the hole and the excitons can also be easily incorporated in the formalism used. Some particular cases are worked out explicitly.