## 2 Formulation of the Problem and Proposed Solution
### Proposed Solution  
By virtue of Helmholtz decomposition, the surface current density on the disk can be thought of as the superposition of a surface curl-free contribution and a surface divergence-free contribution [32]. As a result, the \(n\)-th azimuthal harmonic of the surface current density can be rewritten as follows:  
\[\mathbf{\bar{J}}^{(n)}(\rho)=\underbrace{\left(\frac{d/d\rho}{n/\rho}\right) \Phi_{\mathrm{C}}^{(n)}(\rho)-j \begin{pmatrix}n/\rho\\ d/d\rho\end{pmatrix} \Phi_{\mathrm{D}}^{(n)}(\rho)}_{\mathrm{C}}, \tag{12}\]  
where \(\Phi_{T}^{(n)}(\cdot)\) with \(T=\mathrm{C},\mathrm{D}\) are suitable scalar potential functions. Despite what happens to the cylindrical components of \(\mathbf{\bar{J}}^{(n)}(\rho)\), the VHT\({}_{n}\) of \(\mathbf{\bar{J}}_{T}^{(n)}(\rho)\) are scalar functions [33], i.e.,  
\[ \begin{pmatrix}\overset{\sim}{J}_{\mathrm{C}}^{(n)}(w)\\ 0\end{pmatrix} =\int\limits_{0}^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho) \mathbf{\bar{J}}_{\mathrm{C}}^{(n)}(\rho)\rho d\rho, \tag{13a}\] \[ \begin{pmatrix}0\\ -\mathrm{j}\overset{\sim}{J}_{\mathrm{D}}^{(n)}(w)\end{pmatrix} =\int\limits_{0} ^{+\infty}\underline{\mathbf{H}}^{(n)}(w\rho)\mathbf{\bar{J}}_{\mathrm{D}}^{( n)}(\rho)\rho d\rho, \tag{13b}\]  
or, equivalently,  
\[\overset{\sim}{J}_{T}^{(n)}(w)=\int_{0}^{+\infty}J_{n}(w\rho)\Phi_{T}^{(n)}( \rho)w\rho d\rho, \tag{14}\]  
thus suggesting \(\mathbf{\bar{J}}_{T}^{(n)}(\rho)\) as new unknowns in the spatial domain in order to manage scalar unknowns in the spectral domain. The discretization of the integral equation in (10) is provided by using the [PERSON] method. For this purpose, the following complete and non-redundant expansions of the scalar unknowns in the spectral domain of the orthonormal eigenfunctions of the most singular part of the integral operator that allow the reconstruction of the physical behavior of the surface current density are considered [34]:  
\[\overset{\sim}{J}_{T}^{(n)}(w)=\sum\limits_{h=-1+\delta_{\kappa,\theta}}^{+ \infty}\gamma_{T,h}^{(n)}\overset{\sim}{J}_{T}^{(n,h)}(w), \tag{15}\]  
where  
\[\overset{\sim}{J}_{T}^{(n,h)}(w)=\int_{0}^{+\infty}J_{n}(w\rho)f_{T}^{(n,h)}( \rho)w\rho d\rho=\sqrt{2(|n|+2h+p_{T}+1)}\frac{J_{|n|+2h+p_{T}+1}(aw)}{w^{p_{T} }}, \tag{16}\]and where \(f_{T}^{(n,h)}(\rho)\) denote the expansion functions of \(\Phi_{T}^{(n)}(\cdot)\), \(p_{\mathrm{C}}=3/2\) and \(p_{\mathrm{D}}=1/2\) in order to reconstruct the appropriate edge behavior of the surface current density, and the condition  
\[\gamma_{\mathrm{D},-1}^{(n)}=\alpha^{(n)}\gamma_{\mathrm{C},-1}^{(n)}, \tag{17}\]  
where \(\alpha^{(n)}=j\frac{a\ \mathrm{sgn}(n)}{\sqrt{4n^{2}-1}}\), has to be verified to guarantee the vanishing of the reconstructed current for \(\rho>a\). Simple algebraic manipulations, the [PERSON] method, the orthogonality property of the azimuthal harmonics, and the extraction of the most singular part of the integral operator allow the recasting of the integral equation in (10) as the following second-kind matrix equation:  
\[\mathbf{x}^{(n)}+\mathbf{A}^{(n)}\mathbf{x}^{(n)}=\mathbf{c}^{(n)}, \tag{18}\]  
where, by setting \(\hat{\alpha}^{(n)}=\frac{1}{\sqrt{1+\left(k_{2}\alpha^{(n)}\right)^{2}}}\), the unknown vector is  
\[\mathbf{x}^{(n)}= \begin{bmatrix}\frac{1}{\sqrt{2\omega\
u_{2}}}\mathbf{x}_{ \mathrm{C}}^{(n)}\\ \sqrt{\frac{\omega\
u_{2}}{2}}\mathbf{x}_{\mathrm{D}}^{(n)}\end{bmatrix} , \tag{19a}\] \[\mathbf{x}_{\mathrm{C}}^{(n)}=\left\{x_{\mathrm{C}}^{(n)}\right\}_{h=-1+\delta _{n,0}}^{+\infty},\] (19b) \[\mathbf{x}_{\mathrm{D}}^{(n)}=\left\{x_{\mathrm{D},h}^{(n)}\right\}_{h=0}^{+\infty},\] (19c) \[x_{\mathrm{C},-1}^{(n)}=\frac{\gamma_{\mathrm{C},-1}^{(n)}}{\hat{\alpha}^{(n) }}\ \ \mathrm{for}\ \ \ n\
eq 0,\] (19d) \[x_{\mathrm{T},h}^{(n)}=\gamma_{\mathrm{T},h}^{(n)}\ \ \mathrm{for}\ \ \ h>0, \tag{19e}\]  
the coefficient matrix is  
\[\mathbf{A}^{(n)}=