From Dieudonné's "History of Functional Analysis" I learned that Picard in 1893 gave a characterization of an eigenvalue of the Laplacian as the simple pole of a meromorphic function.

Is there an earlier source that makes this link?

And who named this meromorphic function the resolvent? I have read somewhere that it was Hilbert.

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    $\begingroup$ It is always hard to prove that someone did something for the first time, but in this case it seems you are right: it was Picard. $\endgroup$ Feb 26 '17 at 19:33
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    $\begingroup$ One candidate I have thought of is Cauchy who certainly knew enough about complex analysis and eigenvalue theory to make this connection, but in his 1829 paper on the latter subject he treated eigenvalues as roots of the characteristic equation. $\endgroup$ Feb 27 '17 at 17:27
  • $\begingroup$ Yes, it was Hilbert who coined the term resolvent, in his "Fourth Communication on Integral Equations" gdz.sub.uni-goettingen.de/id/… $\endgroup$ Jan 17 '19 at 19:29

A paper of Kneser (1904) strongly suggests that the idea does (indeed) go back to Cauchy, in connection with Sturm-Liouville problems (i.e. ordinary differential operators, as opposed to the Laplacian in the body of your question). Given functions $g,k,l$ and writing $\smash{L=\frac d{dx}\left(k\frac{d}{dx}\,\cdot\right) - l},$ Kneser considers the [for us: “eigenvalue”] problem $$ LV + rgV=0 $$ with boundary conditions $$ \left[k\frac{dV}{dx}-hV\right](0)=0,\qquad \left[k\frac{dV}{dx}+HV\right](X)=0. \tag1 $$ Addressing the Sturm-Liouville (1837) question whether any $f(x)$ can be expanded into a series of solutions $V_\nu$ belonging to [“eigenvalues”] $r_\nu$: $$ f(x) = A_1V_1+A_2V_2+\cdots, \tag3 $$ Kneser writes:

The particular analytic developments I use for this are inspired or drawn from relevant work of Dini (1880), Harnack (1887), Poincaré (1894, 1895) and Stekloff (1901); but the basic idea can be explained as follows.

These recent authors all use a device introduced by Cauchy (1827) in his study of Fourier series: they build a function of a complex variable $r$ containing $x$ as a parameter, having poles at $r=r_\nu$ as its only singularities, and producing as residues the corresponding terms of the series $(3)$. Poincaré, apparently, first pointed out [I guess here: (1894, 1895)] that Cauchy's auxiliary function is the solution of the equation [of the “resolvent” $\smash{(L+rg)^{-1}}$] $$ LV+rgV+f(x)=0 $$ satisfying conditions $(1)$.

While this early literature is no easy reading, the treatises of Picard (1893, pp. 167-183), Poincaré (1895, pp. 210-223) and Watson (1922, pp. 576-617) have chapter-long expositions of what may be the first three cases historically:

  1. $(g,k,l)=(1,1,0)$ on $[0,\pi]$ with Dirichlet boundary conditions. Then $V_\nu=\sin(\nu x)$, $\smash{r_\nu=\nu^2}$, and (3) is a Fourier sine series. Or Neumann conditions, $V_\nu=\cos(\nu x)$, and cosine series; which Picard (p. 177) and Poincaré (p. 220) attribute to Cauchy (1827, pp. 364-365).

  2. $(g,k,l)=(1,1,0)$ on $[0,1]$ with Fourier's sphere cooling conditions (1822, pp. 340-342): $V(0)=0$ and $V(1)=AV'(1)$ for some $A>1$. Then $V_\nu=\sin(k_\nu x)$ and $\smash{r_\nu=k_\nu^2}$ where the $k_\nu$ are the positive solutions of $\tan(k)=Ak$. So (3) is a “nonharmonic Fourier series,” which Picard (pp. 178-183) and Poincaré (pp. 168-179, 220-223) attribute to Cauchy.

  3. $(g,k,l)=(x,x,a^2/x)$ on $[0,1]$ with $V(1)=0$, and no condition at the singular endpoint $0$. Then $V_\nu=J_a(k_\nu x)$ and $\smash{r_\nu=k_\nu^2}$ where the $k_\nu$ are the roots of the Bessel function $J_a$. So (3) is a “Fourier-Bessel series,” which Watson (pp. 582-591) attributes to Schläfli (1876).

Note added: An earlier (and perhaps clearest) statement by Cauchy occurs in his Application du calcul des résidus à l’intégration des équations différentielles linéaires et à coefficients constants (Exercices de mathématiques 1 (1826) 202-204 = Œuvres (2) 6 (1887) 252-255):

Consider first the task of integrating the differential equation $$ \frac{d^ny}{dx^n} + a_1 \frac{d^{n-1}y}{dx^{n-1}} + a_2 \frac{d^{n-2}y}{dx^{n-2}} + \ldots + a_{n-1} \frac{dy}{dx} + a_ny=0, \tag1 $$ where $a_1, a_2,\dots a_{n-1}, a_n$ denote constant coefficients; and let, for short $$ F(r) = r^n + a_1r^{n-1} + a_2r^{n-2}+\dots+ a_{n-1}r + a_n. \tag2 $$ It is clear that, to satisfy the equation (1), it will suffice to take $$ y = \raise{-1ex}{\huge{\mathcal E}}\,\frac{\varphi(r)\,e^{rx}}{((\,F(r)\,))}, \tag3 $$ where $\varphi(r)$ denotes any function of $r$ which does not become infinite for values of $r$ that verify the formula $$ F(r)=0. \tag4 $$

(Of course, this isn’t yet framed in terms of eigenvalues of the differential operator. It becomes so if we replace $a_n$ by $a_n -\lambda$, but Cauchy didn’t call (4) “characteristic equation” until 1839, and names for its roots seem to have come even later — I’m not sure when.)

Also, for symmetric operators (or quadratic forms) on $\mathbf R^n$ this is all in Weierstrass (1859), cf. p. 219.


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