发表机构
Central China Normal University（华中师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对横向受限极化子的Pekar泛函，证明小限制强度下基态的对称性相关唯一性，大限制强度下推导能量渐近展开并证明极小值的极限形式与唯一性，为维度约化现象提供严格基态对应。
AI 中文摘要
我们研究带有横向简谐势的Pekar泛函的极小值，该模型描述了极化子在横向平面内受简谐限制但沿纵向自由的状态。对于任意限制强度Ω>0，极小值存在，且沿x₃方向平移后，在(x₁,x₂)内径向递减、在x₃内对称递减。对于小Ω>0，我们证明其在上述对称变换下的唯一性；对于足够大的Ω>0，我们推导基态能量的三项渐近展开，并证明经适当缩放后，极小值在H¹(ℝ³)∩L^∞(ℝ³)中收敛于二维线性简谐振荡器归一化基态与有效一维非线性局部问题基态的乘积，呈现渐近变量分离。这为横向受限Pekar模型提供了三维到一维维度约化现象的严格基态对应，该现象此前已在[W. Z. Bao, H. Y. Jian, N. J. Mauser和Y. Zhang, SIAM J. Appl. Math., 2013]中针对各向异性限制势的库仑型薛定谔方程数值观测到。此外，对于所有足够大的Ω>0，我们也建立了上述对称变换下的唯一性。
英文摘要
We study minimizers of the Pekar functional with a transverse harmonic potential, which models a polaron harmonically trapped in the transverse plane but free along the longitudinal direction. For any confinement strength $Ω>0$, minimizers exist and, up to a translation along the $x_3$-direction, are radially decreasing in $(x_1,x_2)$ and symmetric decreasing in $x_3$. For small $Ω>0$ we prove uniqueness up to these symmetries. For sufficiently large $Ω>0$ we derive a three-term asymptotic expansion of the ground state energy and show that, after a suitable rescaling, the minimizers converge in $H^1(\mathbb{R}^3)\cap L^\infty(\mathbb{R}^3)$ to the product of the normalized ground state of a two-dimensional linear harmonic oscillator and that of an effective one-dimensional nonlinear local problem, displaying asymptotic variable separation. This gives, for the transversely confined Pekar model, a rigorous ground-state counterpart of the three-dimensional-to-one-dimensional dimension-reduction phenomenon numerically observed for Coulomb-type Schrödinger equations with anisotropic confining potentials in [W. Z. Bao, H. Y. Jian, N. J. Mauser and Y. Zhang, SIAM J. Appl. Math., 2013]. Moreover, uniqueness up to the same symmetries is also established for all sufficiently large $Ω>0$.
