A Dirac Delta Operator

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 at T. When T is a bounded operator, then is an operator-valued distribution. If T is unbounded, is a more general object that still retains some properties of distributions. We provide an explicit representation of in some particular cases, derive various operative formulas involving and give several applications of its usage in Spectral Theory as well as in Quantum Mechanics.