Ground States of a Transversely Confined Polaron: Uniqueness and an Explicit Limiting Profile
Abstract
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 x3-direction, are radially decreasing in (x1,x2) and symmetric decreasing in x3. 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 H1(ℝ3)∩ L∞(ℝ3) 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.
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BibTeX
@misc{wu2026ground,
title = {{Ground States of a Transversely Confined Polaron: Uniqueness and an Explicit Limiting Profile}},
author = {Shuang Wu and Hongfei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11936},
eprint = {2610.11936},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11936},
}
RIS
TY - UNPB
TI - Ground States of a Transversely Confined Polaron: Uniqueness and an Explicit Limiting Profile
AU - Shuang Wu
AU - Hongfei Zhang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11936
UR - https://arxiv.org/abs/2610.11936
AB - 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(ℝ^3)∩ L^∞(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.
ER -