On the complete solution of the Sturm-Liouville problem (d2X/dx2)+λ2X=0 over a closed interval

We discuss the sets of orthogonal functions that form solutions to the Sturm-Liouville problem for the equation d2X/dx2+λ2X=0 and for the general unmixed boundary conditions over a closed interval of the variable x. The conditions for the presence of the solutions with λ2≤0 are specifically considered and their necessity for completeness of a set of eigenfunctions. Their implications are discussed for three examples from mathematical physics, showing that although for some problems the solutions, corresponding to the negative values of λ2, may reflect physically unusual boundary conditions, their presence is necessary in the general solution for the drift diffusion equation where they may represent stationary or growing in time solutions