A primer on eigenvalue problems of non-self-adjoint operators

Non-self adjoint operators describe problems in science and engineering that lack symmetry and unitarity. They have applications in convection-diffusion processes, quantum mechanics, fluid mechanics, optics, wave-guide theory, and other fields of physics. This paper reviews some important aspects of the eigenvalue problems of non-self-adjoint differential operators and discusses the spectral properties of various non-self-adjoint differential operators. Their eigenvalues can be computed for ground and perturbed states by their spectra and pseudospectra. This work also discusses the contemporary results on the finite number of eigenvalues of non-self-adjoint operators and the implications it brings in modeling physical problems.