A Dirac delta operator

TL;DR

This paper introduces a novel operator-valued delta function at a self-adjoint operator T, extending distribution concepts to unbounded operators with various formulas and applications.

Contribution

It defines and analyzes the delta function operator at a self-adjoint operator, extending distribution theory to unbounded operators with new formulas and applications.

Findings

Defined the delta function operator for bounded and unbounded T.

Derived operative formulas involving the delta operator.

Presented applications demonstrating the operator's utility.

Abstract

If $T$ is a (densely defined) self-adjoint operator acting on a complex Hilbert space $\mathcal{H}$ and $I$ stands for the identity operator, we introduce the delta function operator $\lambda \mapsto \delta \left(\lambda I-T\right) $ at $T$. When $T$ is a bounded operator, then $\delta \left(\lambda I-T\right) $ is an operator-valued distribution. If $T$ is unbounded, $\delta \left(\lambda I-T\right) $ is a more general object that still retains some properties of distributions. We derive various operative formulas involving $\delta \left(\lambda I-T\right) $ and give several applications of its usage.