The critical 2D delta-Bose gas as mixed-order asymptotics of planar Brownian motion

TL;DR

This paper analyzes the 2D delta-Bose gas with a focus on critical coupling, establishing convergence of approximate semigroups for two particles and extending results to multiple particles, using advanced probabilistic methods.

Contribution

It introduces new methods applying excursion theory and ergodicity to prove convergence of semigroups in the critical 2D delta-Bose gas, extending previous functional analytic results.

Findings

Convergence of approximate semigroups for two particles at the critical coupling.

Extension of convergence results to N-particle systems for all N ≥ 3.

Application of excursion theory and winding number ergodicity in the analysis.

Abstract

We consider the 2D delta-Bose gas by a smooth mollification of the delta potential, where the coupling constant is in the critical window. The main result proves that for two particles, the approximate semigroups on $\mathscr B_b(\Bbb R^2)$ for the Schr\"odinger operator with singular interaction at the origin converge pointwise in the initial condition. This convergence extends earlier functional analytic results for the convergence in the $L_2$-norm resolvent sense. The central methods introduced here apply the excursion theory of the 2D Bessel process and the ergodicity of the winding number of planar Brownian motion. The limiting semigroup thus shows both the Kallianpur--Robbins law for additive functionals of planar Brownian motion and Kasahara's second-order law for the fluctuations. As an application, the mode of convergence is extended to the $N$-particle delta-Bose gas for all $N\geq 3$.