real functions $\xi(\vec{x})$ belong to the Hilbert space of square integrable vector functions, $\xi(\vec{x})\in \oplus_{a=1}^{4}\, L^2_a(\mathbb{R}^2)$. A previous step to quantize this system is to perform the normal mode expansion, i.e., use the eigenvectors of ${\cal H}^+$ as a base to expand the fluctuations: $${\cal H}^+ \xi_\omega(\vec{x})=\omega^2 \xi_\omega(\vec{x}) \, \, , \, \, \omega^2\geq 0 \quad , \qquad \quad \xi(\vec{x},t)= \int[d\omega] \, e^{i \omega t} a^t(\omega) \, \xi_\omega(\vec{x}) \label{fluctspec} \, \, .$$ It is well known, see [@Alonso2016:jhep] and References quoted therein to find a summary, that the ${\cal H}^+$ operator has a kernel of dimension $2 N$, i.e., there are $2 N$ lineally independent eigenfunctions of zero eigenvalue in the spectrum of ${\cal H}^+$. There is also a discrete set of eigenfunctions with positive eigenvalues but lesser than one: $0<\omega^2<1$. These are eigenfunctions of ${\cal H}^+$ where the positive fluctuations are trapped in bound states at the vortex core. Finally, there are eigenfunctions in the continuous spectrum of ${\cal H}^+$ with threshold precisely at $\omega^2=1$. In formula (\[fluctspec\]) the $a^t(\omega)$-coefficients describe the four-vector normal modes of fluctuation and the integration symbol $\int [d\omega]$ means that the expansion encompasses both fluctuations in the pure point spectrum and those in the 