Analytic properties of the structure function for the one-dimensional one-component log-gas

Abstract

The structure function $S(k;\beta )$ for the one-dimensional one-component log-gas is the Fourier transform of the charge-charge, or equivalently the density-density, correlation function. We show that for $|k|<\min (2\pi \rho ,2\pi \rho \beta )$, $S(k;\beta )$ is simply related to an analytic function $f(k;\beta )$ and this function satisfies the functional equation $f(k;\beta )=f(-2k/\beta ;4/\beta )$. It is conjectured that the coefficient of $k^j$ in the power series expansion of $f(k;\beta )$ about $k=0$ is of the form of a polynomial in $\beta /2$ of degree $j$ divided by $(\beta /2{)}^j$. The bulk of the paper is concerned with calculating these polynomials explicitly up to and including those of degree 9. It is remarked that the small $k$ expansion of $S(k;\beta )$ for the two-dimensional one-component plasma shares some properties in common with those of the one-dimensional one-component log-gas, but these break down at order $k^8$.

31 pages

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