Abstract
In the Hilbert space, we consider the minimal third order degenerate differential operator L0 with unbounded second order intermediate coefficient r. L0 is not the semibounded operator: its domain is not subspace of some Sobolev space. L0 type operators appear in the physics of the solid and atmosphere and in the electrohydrodynamics. In addition, this operator describes the propagation of a radial pulse wave and processes occurring in an environment capable of keeping information, and it is found in some spectral problems. We assume that |r| strongly increases at infinity and weakly oscillates. Using method of separability theory and the weighted Hardy type inequalities, we find the sufficient conditions to r, under which L0 is closable, and its closure L is the bounded invertible and separable operator.