## 2 Super-Operator Integral Equation for Scattering of Quantum Light by a Dielectric Body  
### Preliminaries  
The super-operator equation in our case may be written as  
\[(\bar{\mathbf{I}}-\
u\overline{\mathbf{A}})\hat{x}=\hat{f}, \tag{1}\]  
where \(\
u\) is a spectral parameter (\(c\)-number), \(\mathbf{I}\) is the unit super-operator, and \(\hat{x}\) and \(\hat{f}\) are operators in the space of quantum states (\(\hat{x}\) is unknown, while \(\hat{f}\) is given). The formal solution of Equation (1) is \(\hat{x}=\overline{\mathbf{R}}(\
u)\hat{f}=(\bar{\mathbf{I}}-\
u\overline{ \mathbf{A}})^{-1}\hat{f}\), where \(\overline{\mathbf{R}}(\
u)\) is the resolvent, which in our case is a super-operator too. As mentioned above, the operators \(\hat{x}\) and \(\hat{f}\) must satisfy the required commutation relations. Therefore, we encounter the main difference between the super-operator equations of the type (1) and their classical operator analogs. In contrast to the classical case, the resolvent \(\overline{\mathbf{R}}(\
u)\) acts not on the whole Hilbert space, but on the subspace limited by the commutation relations. As will be shown below, the action of the resolvent is able to remove the operator \(\hat{x}\) out of the defined subspace. As a result, one may obtain a mathematically exact, but physically incorrect solution with inadequate commutation properties. In other words, Equation (1) is related to ill-posed mathematical problems and should be regularized for its correct application. Its regularization will be given in this paper. It is based on the introduction of the ancillary components, which are associated with a special type of quantum noise. The configuration under consideration is shown in Figure 1. We decompose the operator of the longitudinal electric field into positive and negative frequency components [38; 39] as \(\hat{E}(\mathbf{x},t)=\hat{E}^{+}(\mathbf{x},t)+\hat{E}^{-}(\mathbf{x},t)\) and present it in the frequency domain as  
\[\hat{E}^{+}(\mathbf{x},t)=i\int\limits_{0}^{\infty}\hat{U}_{k}(\mathbf{x})e^{- i\omega_{k}t}dk, \tag{2}\]  
where \(\omega_{k}=kc\), \(k\) is wavenumber, and \(c\) is light velocity in free space. A similar representation for \(\hat{E}^{-}(\mathbf{x},t)\) is obtained from Equation (2) via its Hermitian conjugation. It will be useful to introduce the vector potential \(\hat{A}(\mathbf{x},t)=\hat{A}^{+}(\mathbf{x},t)+\hat{A}^{-}(\mathbf{x},t)\) which is coupled with the field by the relation \(\hat{E}(\mathbf{x},t)=-\partial_{t}\hat{A}(\mathbf{x},t)\). For the spectral density, we have the Helmholtz equation  
\[\
abla_{\perp}^{2}\hat{U}_{k}+k^{2}\varepsilon(\mathbf{x})\hat{U}_{k}=0, \tag{3}\]  
where the symbol \(\perp\) designates two-dimensional Laplacian and  
\[\varepsilon(\mathbf{x})=\left\{ \begin{aligned} &\varepsilon,\mathbf{x}\in S \\ & 1,\mathbf{x}\
otin S.\end{aligned} \right. \tag{4}\]  
The spectral density operator \(\hat{U}_{k}(\mathbf{x})\) plays a role in the complex amplitude of the classical case. Like in the classical case, it satisfies boundary conditions at the boundary of the body (continuity of the tangential components of the field) and radiation condition. Following standard techniques of classical electrodynamics, one can write the solution in terms of the resolvent. For the positive frequency component of the field inside the body, the solution reads  
\[\hat{U}_{k}(\mathbf{x})=\hat{U}_{k0}(\mathbf{x})-k^{2}(\varepsilon-1)\int \limits_{S}\Gamma\big{(}\mathbf{x},\mathbf{x}^{\prime};k\big{)}\hat{U}_{k0} (\mathbf{x}^{\prime})d\mathbf{x}^{\prime}, \tag{5}\]  
where \(\Gamma\big{(}\mathbf{x},\mathbf{x}^{\prime};\
u\big{)}\) is the resolvent kernel defined by Equation (A5). The negative frequency component \(\hat{U}_{k}^{+}(\mathbf{x})\) inside the body may be obtained via the Hermitian conjugation of Equation (5). It is generally accepted to write the resolvent as a function of the spectral parameter \(\
u=-k^{2}(\varepsilon-1)\) as in Equation (A1).