Conformable fractional Bohr Hamiltonian with Bonatsos and double-well sextic potentials
Mohamed Hammad
Abstract
Abstract Using the conformable fractional calculus, a new formulation of the Bohr Hamiltonian is introduced. The conformable fractional energy spectra of free- and two- parameters anharmonic oscillator potentials are investigated. The energy eigenvalues and wave functions are calculated utilizing the finite-difference discretization method. It is proved that the conformable fractional spectra of the free-parameter Bonatsos potentials, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="true">/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:math> close completely the gaps between the classical spectra of the vibrational <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>5</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> dynamical symmetry, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>E</mml:mi> <mml:mrow> <mml:mfenced close=")" open="(" separators=""> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mo>−</mml:mo> <mml:msup> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> models, and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>E</mml:mi> <mml:mrow> <mml:mfenced close=")" open="(" separators=""> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> critical point symmetry. The ground effective sextic potential, which generates both the ground state and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> excited states <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>,</mml:mo> </mml:math> is considered to have two degenerate minima. In this case, the conformable fractional spectra of sextic potentials show a change, as a function of barrier height, from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>γ</mml:mi> </mml:math> -unstable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>O</mml:mi> <mml:mrow> <mml:mfenced close=")" open="(" separators=""> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> energy level sequence to the spectrum of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>E</mml:mi> <mml:mrow> <mml:mfenced close=")" open="(" separators=""> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mo>–</mml:mo> <mml:msup> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msup> </mml:math> model and simultaneously provide new features. The shape coexistence phenomena in the ground band states are identified. The energy spectrum and shape coexistence with mixing phenomena in 96 Mo nucleus are discussed in the framework of the conformable fractional Bohr Hamiltonian.