## Abstract

We analyse the perturbatively calculated adsorption potential $U(U= 2\omega_pz/~\beta$) of a neutral molecule a distance z from a flat metal surface, adopting for the metal the simplest non-trivial spatially dispersive model, namely a continuous fluid confined by impenetrable barriers and with assigned plasma frequency Wp and sound velocity ~$\beta$. Then U emerges as a weighted sum, over molecular states with virtual excitations $E_{n0} = \omega_p\epsilon_{n0}, g(\epsilon_{n0}, y)$, of a function $G(\epsilon_{n0},/~mu)$ which is the Laplace transform of a function $g(\epsilon_{n0}, y)$, where $k = \omega_py/~\beta$ is the tangential component of a plasmon wavenumber. For large $\mu$, Watson's lemma applies because the Taylor series for g converges for small y; the lemma yields an asymptotic series for $G(e,\mu)$ which in general has a leading term of order $\mu^3$. For positive $\epsilon$, this expansion was previously obtained by Mahanty & Paranjape (1977). It must be modified if $\epsilon$ is negative, and in particular if $\epsilon$ is close to the surface-plasmon branch point at $-\frac{1}{\sqrt~{2}}$;~ when $\mu$~ is large but; $\mu|\epsilon + \frac{1}{\sqrt{2}}|$~ is not, the effective leading term is of order $\mu^{-2}$. If e is close to the bulk-plasmon branch point at -1, similar modifications are needed in some next-toleading terms. G is singular at $\mu$ = 0, where its behaviour is governed by the fact that, for large y, g has a convergent expansion in powers of $y^{-1}$. For the Laplace transform G of such a function we prove a theorem which is a natural complement to Watson's lemma but which, surprisingly, seems to be new. It allows G to be expressed in terms of power series in u converging everywhere, with all singularities displayed explicitly: $G = A-{_1}(\epsilon) \mu^{-1}+\Sigma^\infty_{n=0} A_n(\epsilon) \mu^n + ln \Sigma^\infty{n=0} B_n(\epsilon) \mu^n$.~~~ The coefficients $A_{-1}$ and $B_n$ are determined by the coefficients in the expansion of $g(\epsilon, y)$; remarkably, they are entire functions of $\epsilon$, whence those parts of G which are singular in u are entire in $\epsilon$ and vice versa. Sum rules and closure relations allow the sum over molecular states to be found in closed form for the parts of G proportional to $\mu^{-1}$ and ln $\mu$. In the special case where all $\epsilon_{n0}$ are negligible $(\omega_p \gg |E_{n0}|), U(\mu)$ reduces to the far simpler yet still nontrivial classical limit $U_c(/~mu)$, expressible in closed form in terms of Bessel and Struve functions. The non-dispersive limit ($beta$ = 0) is obtainable trivially, but is so special a case that it can shed no light on the small-distance behaviour of U, nor on the large-distance behaviour when e is close to the branch point at $-\frac{1}{\sqrt{2}}$ 'Schwieriger als die Aufstellung der allgemeinen Formeln erweist sich bier wie so haufig ihre spezielle Diskussion.' (Sommerfeld 1909).

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