A comparison theorem for selfadjoint operators

Abstract

In this work we shall establish a result concerning the spectral theory of differential operators. Let L 1 {L_1} and L 2 {L_2} be two self-adjoint operators acting in two different Hubert spaces. Then under some conditions we shall prove that \[ ( d Γ 1 / d Γ 2 ) ( L 2 ) = V ¯ V ′ , (d{\Gamma _1}/d{\Gamma _2})({L_2}) = \overline V V’, \] where Γ 1 ( λ ) {\Gamma _1}(\lambda ) and Γ 2 ( λ ) {\Gamma _2}(\lambda ) are the spectral functions associated with L 1 {L_1} and L 2 {L_2} respectively. V V is the shift operator mapping the set of generalized eigenfunctions of L 1 {L_1} into the set of generalized eigenfunctions of L 2 {L_2} , that is \[ y = V φ , y = V\varphi , \] where L 2 y = λ y {L_2}y = \lambda y and L 1 φ = λ φ {L_1}\varphi = \lambda \varphi .