Partial-range completeness and existence of solutions to two-way diffusion equations

Separating variables to solve a two-way diffusion equation leads to a nonstandard eigenvalue problem in one variable. It is shown that the eigenfunctions having negative eigenvalues are complete on the part of the domain where initial conditions are imposed, while those with positive eigenvalues are complete where final conditions are imposed. The corresponding exponentially growing and exponentially decaying solutions may be used to expand arbitrary solutions on semi-infinite intervals in the ’’time’’ variable. A natural iterative procedure for obtaining solutions on finite intervals is shown to converge. In some cases a linearly growing solution must also be taken into account.