JournalsjstVol. 6 , No. 4pp. 1021–1045

On a coefficient in trace formulas for Wiener–Hopf operators

  • Alexander V. Sobolev

    University College London, UK
On a coefficient in trace formulas for Wiener–Hopf operators cover
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Abstract

Let a=a(ξ),ξR,a = a(\xi), \xi \in \mathbb R, be a smooth function quickly decreasing at infinity. For the Wiener–Hopf operator W(a)W(a) with the symbol aa, and a smooth function g ⁣:CCg\colon \mathbb C \to \mathbb C, H. Widom in 1982 established the following trace formula:

tr(g(W(a))W(ga))=B(a;g),\mathrm {tr}(g(W(a)) - W(g \circ a)) = \mathcal B(a; g),

where B(a;g)\mathcal B(a; g) is given explicitly in terms of the functions aa and gg. The paper analyses the coefficient B(a;g)\mathcal B (a; g) for a class of non-smooth functions gg assuming that aa is real-valued. A representative example of one such function is g(t)=tγg(t) = |t|^{\gamma} with some γ(0,1]\gamma \in (0, 1].

Cite this article

Alexander V. Sobolev, On a coefficient in trace formulas for Wiener–Hopf operators. J. Spectr. Theory 6 (2016), no. 4 pp. 1021–1045

DOI 10.4171/JST/151