The Dirac Delta as a Singular Potential for the 2D Schrodinger Equation

Abstract: In the framework of distributionally generalized quantum theory, the object $H\psi$ is defined as a distribution. The mathematical significance is a mild generalization for the theory of para- and pseudo-differential operators (as well as a generalization of the weak eigenvalue problem), where the $\psi$-do symbol (which is not a proper linear operator in this generalized case) can have its coefficient functions take on singular distributional values. Here, a distribution is said to be singular if it is not L$^p(\mathbb{R}^d)$ for any $p\geq 1$. Physically, the significance is a mathematically rigorous method, which does not rely upon renormalization or regularization of any kind, while producing bound state energy results in agreement with the literature. In addition, another benefit is that the method does not rely upon self-adjoint extensions of the Laplace operator. This is important when the theory is applied to non-Schrodinger systems, as is the case for the Dirac equation and a necessary property of any finite rigorous version of quantum field theory. The distributional interpretation resolves the need to evaluate a wave function at a point where it fails to be defined. For $d=2$, this occurs as $K_o(a|x|)\delta(x)$, where $K_o$ is the zeroth order MacDonald function. Finally, there is also the identification of a missing anomalous length scale, owing to the scale invariance of the formal symbol(ic) Hamiltonian, as well as the common identity for the logarithmic function, with $a,\,b\in\mathbb{R}^+$, $\log(ab)=\log(a)+\log(b)$, which loses unitlessness in its arguments. Consequently, the energy or point spectrum is generalized as a family (set indexed by the continuum) of would-be spectral values, called the C-spectrum.