The One‐Dimensional Coulomb Hamiltonian: Properties of Its Birman–Schwinger Operator

The objective of the present paper is to study in detail the properties of the Birman–Schwinger operator for a self-adjoint realization of the one-dimensional Hamiltonian with the Coulomb potential, both when the Hamiltonian is defined only on (Formula presented.) and when it is defined on the whole real line. In both cases, we rigorously show that the Birman–Schwinger operator is Hilbert–Schmidt, even though it is not trace class, a result that, to the best of our knowledge, has not yet been achieved in the existing mathematically oriented literature on the one-dimensional hydrogen atom. Furthermore, in both cases, we have considered some Hamiltonians depending on a positive parameter approximating the Hamiltonian with the Coulomb potential and proved the convergence of the Birman–Schwinger operators of such approximations to the corresponding one of the Hamiltonian with the Coulomb potential as the parameter goes to zero. The latter result implies the norm resolvent convergence of the approximating Hamiltonians to that of the Hamiltonian with the Coulomb potential.