2023/08/08 by Michael Maroun, Maroun, Michael
Physics and Astronomy · #35Q40 (Primary) 47B93 #8108 #81P68 (Secondary) #81V19 #Analysis of PDEs (math.AP) #Crystallography and Radiation Phenomena #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Quantum Mechanics and Applications #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2308.05195
openalex publication_date 2023/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L2 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 ℝ2), 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.