@Arjun The momentum operator is self-adjoint on the dense subspace of wavefunctions with $\psi(-a) = \psi(a)$ - by their very nature, you can never expect an unbounded operator to be self-adjoint on the entire space, an unbounded
everywhere-defined self-adjoint operator does not exist. For the specific case of the particle on an interval, see
this answer of mine for a discussion of the domains of definition involved