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 <img src=image/13422917_01.gif> at T. When T is a bounded operator, then <img src=image/13422917_02.gif> is an operator-valued distribution. If T is unbounded, <img src=image/13422917_02.gif> is a more general object that still retains some properties of distributions. We provide an explicit representation of <img src=image/13422917_02.gif> in some particular cases, derive various operative formulas involving <img src=image/13422917_02.gif> and give several applications of its usage in Spectral Theory as well as in Quantum Mechanics.
Highlights
For each f ∈ C (R), i. e., for each real-valued continuous function f (λ)
If T is a self-adjoint operator acting on a complex Hilbert space H and I stands for the identity operator, we introduce the delta function operator λ → δ at T
We provide an explicit representation of δ in some particular cases, derive various operative formulas involving δ and give several applications of its usage in Spectral Theory as well as in Quantum Mechanics
Summary
Centro de Investigacion Operativa, Universidad Miguel Hernandez, E-03202 Elche, Spain. Received February 1, 2021; Revised March 18, 2021; Accepted April 9, 2021 Cite This Paper in the following Citation Styles (a): [1] Juan Carlos Ferrando, ”A Dirac Delta Operator,” Mathematics and Statistics, Vol., No.2, pp.
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