eldorado.tu-dortmund.de/server/api/core/bitstreams/a168bbe8-51df-4bec-b568-ea1edd55f9a8/content
Freezing limits for Calogero–Moser–Sutherland particle models
the Bessel functions 𝐽𝐵 (𝑘1,𝑘2)
of type 𝐵𝑁 , we have the following limits; see Refs. 6, 7, 9.
Theorem 7.
(1) For all 𝑥, 𝑦 ∈ 𝐶𝐵𝑁 and 𝜈 ≥ 0, locally uniformly,
lim 𝛽→∞
𝐽𝐵 (𝜈⋅𝛽,𝛽)
( √ 𝛽 ⋅ [...] 𝑡
) ⋅ 𝑤𝐴
𝑘 (𝑦) 𝑑𝑦 (6)
with
𝑤𝐴 𝑘 (𝑥) ∶=
∏ 𝑖<𝑗
(𝑥𝑖 − 𝑥𝑗) 2𝑘, 𝛾𝐴 = 𝑘𝑁(𝑁 − 1)∕2, (7)
and
𝑐𝐴 𝑘 ∶=
( ∫ 𝐶𝐴𝑁
𝑒−‖𝑦‖2∕2 ⋅∏ 𝑖<𝑗
(𝑦𝑖 − 𝑦𝑗) 2𝑘 𝑑𝑦
)−1 =
𝑁!
(2𝜋)𝑁∕2 ⋅
𝑁∏ 𝑗=1 [...] Remark 4 for more details. We next turn to the limit 𝑘 → ∞ in the first approach according to [4, 5, 7-9, 35, 43]. If we start
in 0 ∈ 𝐶𝐴𝑁 , we write the densities of 𝑋𝑡,𝑘∕ √ 𝑡𝑘 as
𝑐𝑜𝑛𝑠𝑡.(𝑘) …
