AUTHOR=Calçada Marcos , Lunardi José T. , Manzoni Luiz A. , Monteiro Wagner , Pereira Marciano 

TITLE=A Distributional Approach for the One-Dimensional Hydrogen Atom

JOURNAL=Frontiers in Physics

VOLUME=7

YEAR=2019

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2019.00101

DOI=10.3389/fphy.2019.00101

ISSN=2296-424X

ABSTRACT=<p>We consider the one-dimensional Hydrogen atom, with the Coulomb interaction <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mfrac></mml:math></inline-formula> (γ &lt; 0), and use Schwartz's theory of distributions to address the non-integrable singularity at the origin. This singularity renders the interaction term <italic>V</italic>(<italic>x</italic>)ψ(<italic>x</italic>) in the Schrödinger's equation, where ψ(<italic>x</italic>) is the wave function, an ill-defined product in the ordinary sense. We replace this ill-defined product by a well-defined interaction <italic>distribution</italic>, <italic>S</italic>[ψ, <italic>V</italic>](<italic>x</italic>), and by imposing that it should satisfy some fundamental mathematical and physical requirements, we show that this distribution is defined up to a 4-parameter family of contact interactions, in agreement with the method of self-adjoint extensions. By requiring that the interaction distribution be invariant under parity, we further restrict the 4-parameter family of interactions to the subfamily of all the parity invariant Coulomb interactions. Finally, we present a <italic>systematic</italic> study of the bound states within this subfamily, addressing the frequently debated issues of the multiplicity and parity of the bound states, and the boundedness of the ground state energy.</p>