Information theoretic measures for the Dunkl–Coulomb problem

Abstract Motivated by the conclusion section of Ghazouani and Sboui (2020 J. Phys. A: Math. Theor. 53 035202), we merge together the invariance algebra of the Dunkl–Coulomb problem and the Lie algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtext>so</mml:mtext> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> into a bigger new algebras. Next, we introduce several information theoretic measures of the Dunkl–Coulomb problem: Using the Dunkl transform and the canonical Dunkl transform, we obtain an analytical expression for the momentum wave functions of the Dunkl–Coulomb problem. We give an analytical expression for the expectation values of the position <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and momentum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msup> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> in various bound states of the Dunkl–Coulomb problem. With these, a Heisenberg-type inequality for the three-dimensional linear canonical Dunkl transform is derived and is applied to the bound states of the Dunkl–Coulomb problem.