Asymptotic Distribution of Eigenvalues of Non-Self Adjoint Differential Operators

The study of non-self-adjoint differential operators is a historical issue. Before and until now, most studies of operator theory have been about self-adjoint operators. But non-self-adjoint operators have recently found many applications in other sciences. These operators have received much attention in thermodynamics and quantum mechanics. Because there is no general spectral theory for these operators, it is more difficult to study these operators than the self-adjoint type. The spectrum of these operators is usually unstable and their resolvent is unpredictable. In this paper by use of Weyl’s law for asymptotic distribution of eigenvalues of elliptic self adjoint operators on bounded domain, we consider a second-order non-self-adjoint differential operator and study its distribution of eigenvalues. Our goal is to find an asymptotic formula for the Dirichlet eigenvalue counting function on the open interval (0,1)