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On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential. II. (English) Zbl 1382.82014

Bini, Dario A. (ed.) et al., Large truncated Toeplitz matrices, Toeplitz operators, and related topics. The Albrecht Böttcher anniversary volume. Basel: Birkhäuser/Springer (ISBN 978-3-319-49180-6/hbk; 978-3-319-49182-0/ebook). Operator Theory: Advances and Applications 259, 213-234 (2017).
Summary: In this paper we continue our analysis [3] of the determinant \(\mathrm{det} ({I}- {\gamma}{K}_{s})\), \(\gamma \in (0, 1)\) where \(K_s\) is the trace class operator acting in \(L^{2}(-1, 1)\) with kernel \({K}_{s}(\lambda, \mu) = \frac{{\sin}{s}(\lambda-\mu)}{{\pi}(\lambda-\mu)}\). In [T. Bothner et al., Commun. Math. Phys. 337, No. 3, 1397–1463 (2015; Zbl 1321.82027)] various key asymptotic results were stated and utilized, but without proof: Here we provide the proofs (see Theorem 1.2 and Proposition 1.3 below).
For the entire collection see [Zbl 1367.47005].

MSC:

82B23 Exactly solvable models; Bethe ansatz
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
34E05 Asymptotic expansions of solutions to ordinary differential equations
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
15B05 Toeplitz, Cauchy, and related matrices

Citations:

Zbl 1321.82027
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