A General Existence Theorem and Asymptotics for Non-self-adjoint Sturm-Liouville Problems

We prove a general existence theorem for non-self-adjoint Sturm-Liouville problems and we derive quite general asymptotic formulae for their eigenvalues and the corresponding eigenfunctions. The derived formulae are general, very accurate and remain valid for a large class of complex potentials with singularities. These results are obtained by using He's homotopy perturbation method (HPM) with an auxiliary parameter. It will be shown that this method is easy to use and makes the study of these problems more simple and more efficient. In order to illustrate the theory, interesting asymptotic and numerical results are discussed and presented from a wide range of examples.