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 d(2)X/dx(2)+lambda(2)X=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 lambda(2)less than or equal to0 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 lambda(2), 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. (C) 2002 American Institute of Physics.