Coalescence of Poles into a Branch Cut

To appreciate the non-analyticity of the Green's function, consider how the the isolated poles of a finite string Green's function merge so that they form the branch cut when the string becomes infinitely long. To illustrate the point, start with a string of length $\ell$ which satisfies the Dirichlet boundary conditions at both ends.

$$\frac{d^2g_\lambda }{dx^2}+\lambda g_\lambda = -\delta(x-\xi )\quad 0<x,\xi<\ell$$

$$g_\lambda (0;\xi) =$$ 0

$$g_\lambda (\ell;\xi) = 0 ~~.$$

The Green's function is

$$g_\lambda(x;\xi) = \frac{1 }{\lambda^{1/2} \sin \lambda^{1/2}\ell} \sin \{\lambda^{1/2} x_<\} ~ \sin\{ \lambda^{1/2} (\ell -x_>) \}$$

Figure 4.14: Eigenvalue spectra in the complex $\lambda$ -plane

Observe that in the complex $\lambda$ -plane its poles are isolated and located at

$$\lambda_n=\frac{n^2\pi^2}{\ell^2}~~~~~n=1,2,3,\cdots$$

along the real $\lambda$ -axis. Their separation

$$\triangle \lambda_n = \lambda_{n+1}-\lambda_n$$

$$= \frac{\pi^2}{\ell^2} (2n+1)$$

tends to zero as $\ell \to \infty$ . Thus, as depicted in Figure 4.14, as $\ell \to \infty$ the isolated poles of $g_\lambda(x;\xi)$ coalesce into a continuous barrier, the branch cut, which separates the ``outgoing'' from the ``incoming'' wave numbers $\lambda ^{1/2}$ on the same Riemann sheet of $\lambda ^{1/2}$ .

Remark. How would a change in boundary conditions, from Dirichlet to, say, mixed Dirichlet-Neumann conditions, have altered the coalescence of the poles of the Green's function? It is evident that the positions of these poles depend continuously on the parameters that specify the Dirichlet-Neumann boundary conditions. A change in these boundary conditions would merely have shifted these poles along the real $\lambda$ axis in a continuous way. However, as $\ell \to \infty$ , they still would have coalesced and formed the branch cut across which the limiting Green's function is discontinuous in the $\lambda$ plane