The spheroidal wave operator is a linear elliptic operator of second order with smooth coefficients on the unit sphere . Using angular variables and this operator may be written in the form Here is the aspherical parameter. The authors consider the operator in the Hilbert space with boundary conditions It is proved that the spectral representation for is holomorphic in the aspherical parameter in a neighborhood of the real line. For real , estimates are derived for all eigenvalue gaps uniformly in . The proof of the gap estimates is based on detailed estimates for complex solutions of the Riccati equation. The spectral representation for complex is derived using the theory of slightly non-selfadjoint perturbations.