A Complete Proof of Mueller--Ho Conjecture
Abstract
For s>0, and z∈ℍ={z=x+iy:y>0}, define θ(s;z)=∑m,n∈ℤe−πs(m2y+((mx−n)2)/y) and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-π\big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2π(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respectively. We prove that, up to the modular group, there exist three thresholds 0<αa<αb<αc<1 such that alignednonumber Minimaz∈ℍ, (a,b)∈ℝ2(θ(1;z)+αJ(z;a,b))= (eiπ/3;1/3,1/3), &if α∈[0,αa), (eiπ/3;1/3,1/3) or (eiφαa;1/2,1/2), &if α=αa, (eiφα;1/2,1/2), φα∈(φαa,π/2), &if α∈(αa,αb), (i;1/2,1/2), &if α∈[αb,αc], (iyα;1/2,1/2), yα∈(1,√3], y1=√3, &if α∈(αc,1], endaligned where the complex variable z and vector (a,b), represent the total vortex shapes and relative positions of the two-component Bose gas, respectively. Consequently, this completely proves the conjecture of Mueller and Ho~\cite{Mue2002} (2002) arising in Bose--Einstein condensates.
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BibTeX
@misc{luo2026complete,
title = {{A Complete Proof of Mueller--Ho Conjecture}},
author = {Senping Luo and Juncheng Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12203},
eprint = {2610.12203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12203},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Complete Proof of Mueller--Ho Conjecture
AU - Senping Luo
AU - Juncheng Wei
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12203
UR - https://arxiv.org/abs/2610.12203
AB - For s>0, and z∈ℍ={z=x+iy:y>0}, define θ(s;z)=∑_m,n∈ℤe^−πs(m^2y+((mx−n)^2)/y) and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-π\big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2π(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respectively. We prove that, up to the modular group, there exist three thresholds 0<α_a<α_b<α_c<1 such that alignednonumber Minima_z∈ℍ, (a,b)∈ℝ^2(θ(1;z)+αJ(z;a,b))= (e^iπ/3;1/3,1/3), &if α∈[0,α_a), (e^iπ/3;1/3,1/3) or (e^iφ_α_a;1/2,1/2), &if α=α_a, (e^iφ_α;1/2,1/2), φ_α∈(φ_α_a,π/2), &if α∈(α_a,α_b), (i;1/2,1/2), &if α∈[α_b,α_c], (iy_α;1/2,1/2), y_α∈(1,√3], y_1=√3, &if α∈(α_c,1], endaligned where the complex variable z and vector (a,b), represent the total vortex shapes and relative positions of the two-component Bose gas, respectively. Consequently, this completely proves the conjecture of Mueller and Ho~\cite{Mue2002} (2002) arising in Bose--Einstein condensates.
ER -