Juan Carlos Ferrando (Operations Research Center, University Miguel Hernández of Elche)
Abstract: If T is a (densely defined) self-adjoint operator acting on a complex Hilbert space H and I stands for the identity operator, we introduce the delta function operator λ {mapping} δ (λI – T) at T. When T is a bounded operator, then δ (λI – T) is an operator-valued distribution. If T is unbounded, δ (λI – T) is a more general object that still retains some properties of distributions. We provide an explicit representation of δ (λI – T) in some particular cases, derive various operative formulas involving δ (λI – T) and give several applications of its usage in Spectral Theory as well as in Quantum Mechanics.