Topics on the spectral properties of degenerate non-self-adjoint differential operators

Abstract

Abstract Let ( P u ) ( t ) = − d d t ( ω 2 ( t ) q ( t ) d u ( t ) d t ) $( Pu ) ( t ) =- \frac{d}{dt} ( \omega^{2} ( t ) q ( t ) \frac{du ( t )}{dt} )$ be a degenerate non-self-adjoint operator defined on the space H ℓ = L 2 ( 0 , 1 ) ℓ $H_{\ell} = L^{2} (0,1)^{\ell}$ with Dirichlet-type boundary conditions, where ω ( t ) ∈ C 1 ( 0 , 1 ) $\omega(t)\in C^{1} (0,1)$ is a positive function with further assumptions that will be specified later, and q ( t ) ∈ C 2 ( [ 0 , 1 ] , End C ℓ ) $q(t)\in C^{2} ( [ 0,1 ], \operatorname{End} C^{\ell} )$ is a matrix function. In this article, some spectral characteristics of the operator P are considered. We estimate the resolvent of P and then prove the limit argument theorem. Finally, we find a formula for the distribution of eigenvalues of the operator P acting on H ℓ $H_{\ell}$ .

Keywords

• resolvent
• asymptotic spectrum
• eigenvalues
• non-self-adjoint differential operators