  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.
