<p>This paper investigates the influence of the Dunkl operator on the one-dimensional Bose–Einstein condensate (BEC), modeled by the Gross–Pitaevskii equation. Two fundamental external potentials are considered: the free (null potential) case and the harmonic trapping potential. To analyze the system, we employ a variational Gaussian ansatz with odd parity, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\psi (x) \propto x e^{-Bx^2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∝</mo> <mi>x</mi> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>B</mi> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, which is strictly compatible with the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s = -1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> sector (odd eigenvalue of the reflection operator) and describes the first excited state of the condensate. We derive approximate expressions for the energy and optimal width <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$B_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>, and explore the role of the Wigner deformation constant <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> in shaping the condensate’s density profile, energy minimization, and stability under confinement. This work provides a novel perspective on the interplay between generalized reflection symmetries, weighted measures, and nonlinear quantum phenomena in excited states of BECs.</p>

Bose–Einstein Condensate in the Dunkl Formalism: A Study of an Excited Bound State with a Nodal Gaussian Ansatz

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发表日期：

2026年7月27日

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Salah Zenkhri,
Djamel Eddine Zenkhri

摘要

This paper investigates the influence of the Dunkl operator on the one-dimensional Bose–Einstein condensate (BEC), modeled by the Gross–Pitaevskii equation. Two fundamental external potentials are considered: the free (null potential) case and the harmonic trapping potential. To analyze the system, we employ a variational Gaussian ansatz with odd parity, $\psi (x) \propto x e^{-Bx^2}$ $ψ (x) \propto x e^{- B x^{2}}$ , which is strictly compatible with the $s = -1$ $s = - 1$ sector (odd eigenvalue of the reflection operator) and describes the first excited state of the condensate. We derive approximate expressions for the energy and optimal width $B_0$ $B_{0}$ , and explore the role of the Wigner deformation constant $\theta $ $θ$ in shaping the condensate’s density profile, energy minimization, and stability under confinement. This work provides a novel perspective on the interplay between generalized reflection symmetries, weighted measures, and nonlinear quantum phenomena in excited states of BECs.