Let S(x) be a massless scalar quantum field which lives on the three-dimensional hyperboloid xx= (x⁰)²-(x¹)²-(x²)²-(x³)²=-1. The classical action is assumed to be (\hbar=1=c)(8\pi e²)^-1\int dx g^ik\partial _i S\partial _k S, where e² is the coupling constant, dx is the invariant measure on the de Sitter hyperboloid xx=-1 and g_ik, i,k=1,2,3, is the internal metric on this hyperboloid. Let u be a fixed four-velocity i.e. a fixed unit time-like vector. The field S(u)=(1/4 \pi )\int dx\delta (ux)S(x)is smooth enough to be exponentiated, being an average of the operator valued distribution S(x) over the entire Cauchy surface ux=0. We prove that if 0<e2<\pi , then the state |u\rangle = exp (-iS(u))\mid 0\rangle , where \mid 0\rangle is the Lorentz invariant vacuum state, contains a normalizable eigenstate of the Casimir operator C₁=-(1/2)M_{\mu \nu} M^{\mu \nu} ; M_{\mu \nu} are generators of the proper orthochronous Lorentz group. The eigenvalue is (e²/\pi )(2-(e²/\pi )). This theorem was first proven by the Author in 1992 in his contribution to the Czyz Festschrift, see Erratum Acta Phys. Pol. B 23, 959 (1992). In this paper a completely different proof is given: we derive the partial, differential equation satisfied by the matrix element \langle u\mid  exp (-\sigma C₁)\mid u\rangle, \sigma > 0, and show that the function  exp(z)\cdot (1-z)\cdot  exp [-\sigma z (2-z)], z= e²/ \pi , is an exact solution of this differential equation, recovering thus both the eigenvalue and the probability of occurrence of the bound state. A beautiful integral is calculated as a byproduct.

PACS numbers: 12.20.Ds, 11.10.Jj