Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension

• URL: http://arxiv.org/abs/2107.03785v3
• Date: Tue, 5 Jul 2022 11:46:22 GMT
• Title: Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension
• Authors: Jan Derezi\'nski and B{\l}a\.zej Ruba
• Abstract summary: We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_omega,lambda:=beginbmatrix-fraclambda+omegax&-partial_x.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_{\omega,\lambda}:=\begin{bmatrix}-\frac{\lambda+\omega}{x}&-\partial_x \\ \partial_x & -\frac{\lambda-\omega}{x}\end{bmatrix}$. 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.

Related papers

• Small Circle Expansion for Adjoint QCD$_2$ with Periodic Boundary Conditions [0.0]
  Supersymmetry is found at the adjoint mass-squared $g2 hvee/ (2pi)$, where $hvee$ is the dual Coxeter number of $G$. We generalize our results to other gauge groupsG$, for which supersymmetry is found at the adjoint mass-squared $g2 hvee/ (2pi)$, where $hvee$ is the dual Coxeter number of $G$.
  arXiv  Detail & Related papers  (2024-06-24T19:07:42Z)
• Klein-Gordon oscillators and Bergman spaces [55.2480439325792]
  We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $mathbbR3,1$. The general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic functions on the K"ahler-Einstein manifold $Z_6$.
  arXiv  Detail & Related papers  (2024-05-23T09:20:56Z)
• Vacuum Force and Confinement [65.268245109828]
  We show that confinement of quarks and gluons can be explained by their interaction with the vacuum Abelian gauge field $A_sfvac$.
  arXiv  Detail & Related papers  (2024-02-09T13:42:34Z)
• The Dirac Delta as a Singular Potential for the 2D Schrodinger Equation [0.0]
  In the framework of distributionally generalized quantum theory, the object $Hpsi$ is defined as a distribution. The significance is a mathematically rigorous method, which does not rely upon renormalization or regularization of any kind. The distributional interpretation resolves the need to evaluate a wave function at a point where it fails to be defined.
  arXiv  Detail & Related papers  (2023-12-23T00:43:06Z)
• Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
  We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
  arXiv  Detail & Related papers  (2023-06-21T10:11:28Z)
• Quantum Current and Holographic Categorical Symmetry [62.07387569558919]
  A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
  arXiv  Detail & Related papers  (2023-05-22T11:00:25Z)
• On parametric resonance in the laser action [91.3755431537592]
  We consider the selfconsistent semiclassical Maxwell--Schr"odinger system for the solid state laser. We introduce the corresponding Poincar'e map $P$ and consider the differential $DP(Y0)$ at suitable stationary state $Y0$.
  arXiv  Detail & Related papers  (2022-08-22T09:43:57Z)
• Spectral asymptotics for two-dimensional Dirac operators in thin waveguides [0.0]
  We prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schr"odinger operator on $L1(mathbb R)$ [ mathcalL_e := -fracd2ds2 - frackappa2pi2 ] with a geometrically induced potential. The eigenvalues are shown to be at distance of order $varepsilon$ from the essential spectrum, where $2varepsilon$ is the width
  arXiv  Detail & Related papers  (2022-07-18T15:50:10Z)
• Algebraic Structure of Dirac Hamiltonians in Non-Commutative Phase Space [0.0]
  We study two-dimensional Dirac Hamiltonians with non-commutativity both in coordinates and momenta. We analyze the energy spectrum of some simple models by constructing and studying the representation spaces of the unitary irreducible representations.
  arXiv  Detail & Related papers  (2022-05-02T13:20:52Z)
• Infinite-dimensional analyticity in quantum physics [0.0]
  A study is made, of families of Hamiltonians parameterized over open subsets of Banach spaces. It renders many interesting properties of eigenstates and thermal states analytic functions of the parameter.
  arXiv  Detail & Related papers  (2021-08-23T11:49:10Z)
• Anharmonic oscillator: a solution [77.34726150561087]
  The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
  arXiv  Detail & Related papers  (2020-11-29T22:13:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.