On the complete solution of the Sturm–Liouville problem (d[sup 2]X/dx[sup 2])+λ[sup 2]X=0 over a closed interval

Yuriy Sosov, Constantine E. Theodosiou
2002 Journal of Mathematical Physics  
We discuss the sets of orthogonal functions that form solutions to the Sturm-Liouville problem for the equation d 2 X/dx 2 ϩ 2 Xϭ0 and for the general unmixed boundary conditions over a closed interval of the variable x. The conditions for the presence of the solutions with 2 р0 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
more » ... tions, corresponding to the negative values of 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.
doi:10.1063/1.1459753 fatcat:cy4gglt42nak7auon5bjiqlq2e