Following Coddington-Levinson's book Theory of ordinary differential equations, chapter 7: "Self-adjoint problems on finite intervals", let us consider the eigenvalue problem
$\pi(l):\begin{cases} Lx(t)= lx(t) & t \in [a, b] \\ Ux=0 \end{cases}$
where $Lx=p_0(t)x^{(n)}+p_1(t)x^{(n-1)}+\ldots + p_n(t)x(t)$ (with $p_0(t)\ne 0$, the problem is not singular) and $Ux=0$ stands for the boundary conditions
$U_jx=\sum_{k=1}^n(M_{jk}x^{(k-1)}(a)+N_{jk}x^{(k-1)}(b)),\qquad j=1\ldots n.$
Also let $\pi$ be self-adjoint, meaning that $\int_a^b Lu\overline{v}\, dt=\int_a^bu\overline{Lv}\, dt$ for all $u, v \in C^n$ satisfying boundary conditions $Uu=Uv=0$.
We say that $l\in \mathbb{C}$ is an eigenvalue of $\pi$ if $\pi(l)$ admits non trivial solutions. Coddington-Levinson's theorem 2.1 asserts that all eigenvalues are real and that they have no finite cluster point. What is interesting for this question is the proof: the authors start taking a fundamental system $\{\varphi_j, j=1\ldots n\}$ of solutions of the linear equation $Lx=lx$, observing that each $\varphi_j$ depends analytically on $l$. Then they point out that the generic solution
$x=\sum_{j=1}^nc_j \varphi_j$
of $Lx=lx$ is an eigenvalue of $\pi$ if and only if
$\tag{1} \sum_{j=1}^n c_j U_k\varphi_j=0 \qquad k=1\ldots n, $
which is a system of $n$ homogeneous linear equations in $n$ unknowns $c_1 \ldots c_n$. The determinant $\Delta$ of $(1)$ is an entire function of $l$ and it vanishes exactly at the eigenvalues of $\pi$. At this point the authors finish off their proof, while we proceed to our question.
Question This $\Delta$, being entire and vanishing at eigenvalues, might be regarded as an infinite-dimensional analogue of the characteristic polynomial of a matrix. Is there any relationship between the multiplicity of its zeros and the geometric multiplicity of the corresponding eigenvalues (i.e. the dimension of the associated eigenspaces)?
Thank you.