1D Schrödinger operators with Coulomb-like potentials

Abstract

We study the convergence of 1D Schrödinger operators Hε with the potentials which are regularizations of a class of pseudo-potentials having in particular the formThe limit behaviour of Hε in the norm resolvent topology, as ε → 0, essentially depends on a way of regularization of the Coulomb potential and the existence of zero-energy resonances for δ -like potential. All possible limits are described in terms of point interactions at the origin. As a consequence of the convergence results, different kinds of L ∞ (R)-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.2000 Mathematics Subject Classification. Primary 34L40, 34B09; Secondary 81Q10. Key words and phrases. 1D Schrödinger operator, Coulomb potential, one-dimensional hydrogen atom, δ -potential, scattering problem, penetrability of potential, point interaction.

A Distributional Approach for the One-Dimensional Hydrogen Atom

“…The authors would like to thank the referees for bringing references [12,13,15,16] to our attention.…”

“…This regularization procedure has led to conflicting results in which concerns the degeneracy or not of the spectrum, the existence of even wave functions and whether the associated spectrum is continuous or discrete, and the stability of the model (unboundedness of the ground state energy from below, in the a → 0 limit) [1,6,9,10]. This is, perhaps, unsurprising, since regularization procedures need to be employed with great care and often need additional input, such as symmetry, to yield meaningful results, as is well-known in quantum field theory [11] or in the study of chemical indices [12,13]-see [14,15] for rigorous treatments of the regularized 1D Coulomb potential, and [16] for a general treatment of regularized Sturm-Liouville operators.…”

The bound-state solutions of the one-dimensional hydrogen atom

“…Other approaches to studying the one-dimensional Coulomb potential include the use of symmetry arguments, 20,21 the Laplace transform, 22 and the theory of distributions. 16,[23][24][25][26] The d-dimensional hydrogen atom has been studied also, 27,28 although the problems associated with the even-parity solutions in the d = 1 case were not fully addressed.…”