  Analyzing the point spectrum, i.e. bound state energy eigenvalue, of the
Dirac delta function in two and three dimensions is notoriously difficult
without recourse to regularization or renormalization, typically both. The
reason for this in two dimensions is two fold; 1) the coupling constant,
together with the mass and Planck's constant form an unitless quantity. This
causes there to be a missing anomalous length scale. 2) The immediately obvious
L$^2$ solution is divergent at the origin, where the Dirac Delta potential has
its important point of support as a measure. Due to the uniqueness of the
solution presented here, it is immediate that the linear operator (the two
dimensional Laplace operator on all of $\mathbb{R}^2$), with the specialized
domain constructed here, ensures that the point spectrum has exactly one
element. This element is determined precisely, and a natural mathematically
rigorous resolution to the anomalous length scale arises. In this work, there
is no recourse to renormalization or regularization of any kind.
