Degenerated Boundary Conditions of a Sturm-Liouville Problem with a Potential-Distribution

We describe all degenerate boundary conditions in the homogeneous Sturm-Liouville problem with a Potential-Distribution. We show that for the case y(1)(x, lambda) y(2)([1])(x, lambda), the characteristic determinant is zero if and only if the boundary conditions are falsely periodic boundary conditions; the characteristic determinant is identically a nonzero constant if and only if the boundary conditions are generalized Cauchy conditions. For the case the case y(1)(x, lambda) not equivalent to y(2)([1]) (x, lambda) in which the characteristic determinant is identically zero is impossible and that the only possible degenerate boundary conditions are the Cauchy conditions.