Ground States of a Transversely Confined Polaron: Uniqueness and an Explicit Limiting Profile

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 $\mathrm{\Omega }>0$, 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 $\mathrm{\Omega }>0$ we prove uniqueness up to these symmetries. For sufficiently large $\mathrm{\Omega }>0$ we derive a three-term asymptotic expansion of the ground state energy and show that, after a suitable rescaling, the minimizers converge in $H^1({\mathbb{R}}^3)\cap L^{\mathrm{\infty }}({\mathbb{R}}^3)$ 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 $\mathrm{\Omega }>0$.