Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension

Abstract

We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_{\omega ,\lambda }:=\left[\begin{array}{cc}-\frac{\lambda +\omega }{x}& -{\partial }_x\\ {\partial }_x& -\frac{\lambda -\omega }{x}\end{array}\right]$. We describe their closed realizations in the sense of the Hilbert space $L^2({\mathbb{R}}_{+},{\mathbb{C}}^2)$, allowing for complex values of the parameters $\lambda ,\omega$. In physical situations, $\lambda$ is proportional to the electric charge and $\omega$ is related to the angular momentum. We focus on realizations of $D_{\omega ,\lambda }$ homogeneous of degree $-1$. They can be organized in a single holomorphic family of closed operators parametrized by a certain 2-dimensional complex manifold. We describe the spectrum and the numerical range of these realizations. We give an explicit formula for the integral kernel of their resolvent in terms of Whittaker functions. We also describe their stationary scattering theory, providing formulas for a natural pair of diagonalizing operators and for the scattering operator. It is well-known that $D_{\omega ,\lambda }$ arise after separation of variables of the Dirac-Coulomb operator in dimension 3. We give a simple argument why this is still true in any dimension. Furthermore, we explain the relationship of spherically symmetric Dirac operators with the Dirac operator on the sphere and its eigenproblem. Our work is mainly motivated by a large literature devoted to distinguished self-adjoint realizations of Dirac-Coulomb Hamiltonians. We show that these realizations arise naturally if the holomorphy is taken as the guiding principle. Furthermore, they are infrared attractive fixed points of the scaling action. Beside applications in relativistic quantum mechanics, Dirac-Coulomb Hamiltonians are argued to provide a natural setting for the study of Whittaker (or, equivalently, confluent hypergeometric) functions.

arXiv commentVersion published in Annales Henri Poincaré

References in corpus (13)

• Self-adjointness of Dirac operators via Hardy-Dirac inequalities
• On Schroedinger operators with inverse square potentials on the half-line
• Exact solutions to the Dirac equation for a Coulomb potential in $D+1$ dimensions
• Self-adjoint realisations of the Dirac-Coulomb Hamiltonian for heavy nuclei
• Self-Adjoint Extensions for the Dirac Operator with Coulomb-Type Spherically Symmetric Potentials
• On radial Schroedinger operators with a Coulomb potential
• Discrete spectra for critical Dirac-Coulomb Hamiltonians
• 1-dimensional Schrodinger operators with complex potentials
• Dirac-Coulomb operators with general charge distribution. I. Distinguished extension and min-max formulas
• Self-Adjoint Extensions of Dirac Operator with Coulomb Potential
• Distinguished self-adjoint extension of the two-body Dirac operator with Coulomb interaction
• Boundary triples for the Dirac operator with Coulomb-type spherically symmetric perturbations
• Continuous and holomorphic functions with values in closed operators

Cited by in corpus (1)

• Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors