Asymptotics of Determinants of Bessel Operators

Abstract

For a ∈ L ∞ (R + ) ∩ L 1 (R + ) the truncated Bessel operator B τ (a) is the integral operator acting on L 2 [0, τ ] with the kernelwhere J ν stands for the Bessel function with ν > −1. In this paper we determine the asymptotics of the determinant det(I + B τ (a)) as τ → ∞ for sufficiently smooth functions a for which a(x) = 1 for all x ∈ [0, ∞). The asymptotic formula is of the form det(I + B τ (a)) ∼ G τ E with certain constants G and E, and thus similar to the well-known Szegö-Akhiezer-Kac formula for truncated Wiener-Hopf determinants.

Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain

“…Since we were able to derive an exact determinant formula for it, the return probability is perhaps the simplest observable for which a rigorous long time analysis can be performed. Indeed, similar determinants have been studied using differential equations [78,79], operator theoretic [67,80], or Riemann-Hilbert techniques [55,56,59,81]. While such methods fall outside the scope of the present paper, a proof of the nowhere continuous behavior, together with the exponential decay away from roots of unity is left as a pressing issue.…”

A Brief History of the Strong Szegö Limit Theorem

“…Using the same trick as in the first section to pull out the ( ) term, but with a different approach to computing the infinite determinants one can show the following. For details see [1] and for similar computations see [2,3,4,5] Theorem 2. Let and be a compatible pair, and let ∈ and = exp( ).…”

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“…Suppose that φ 1 , φ 2 ∈ L 2 (0, ∞), and extend them to L 2 (−∞, ∞) by letting φ j (u) = 0 for all u < 0. Then by a simple Fourier transform calculation as in [5] 2α −2α…”