On classification of second-order differential equations with complex coefficients

This paper is concerned with the deficiency index problem of second-order differential equations with complex coefficients. It is known that this class of equations is classified into cases I, II, and III according to the number of linearly independent solutions in suitable weighted square integrable spaces. In this study, the original equation is reformulated into a new formally self-adjoint differential system by introducing a new spectral parameter and the relationship between the classifications of the equation and the system is obtained. Moreover, the exact dependence of cases II and III on the corresponding half planes is given and some criteria of the three cases are established.