1 Introduction

In this paper, which is a follow-up of [1], we consider dielectric radiating cavities [13, 15, 25] and rigorously obtain asymptotic formulas for the shifts in the scattering resonances that are due to a small particle of arbitrary shape. Our formula shows that the perturbations of the scattering resonances are proportional to the polarization tensor of the small particle. Therefore, the shift of the scattering frequencies induced by the small particle is of the order of the particle’s volume. When the particle is excited near its resonant frequencies, its polarization tensor blows up and consequently, as shown in this paper, an anomalous shift of the scattering resonances can be observed when the resonant particle is coupled to the cavity modes. We also consider the case where the scattering resonances are exceptional. Exceptional scattering resonances can be defined as the poles of the Green’s function associated with the radiating cavity which are not simple [2, 3, 12, 20]. They owe their existence to the non-Hermitian character of the scattering resonance problem [12, 20]. Optical cavities that operate at exceptional scattering frequencies can be exploited for enhanced nanoparticle sensing [16, 21]. In this paper, we prove that a small particle inside a cavity perturbs the system from its exceptional points, leading to frequency splitting. Moreover, the splitting induced by the particle is of a much larger amplitude than suggested by the particle’s volume. In fact, we consider exceptional points of order two and derive a formula for the splitting of such resonances induced by a small particle. We prove that the strength of the splitting of the exceptional scattering frequencies is proportional to the square root of the volume of the particle.

Our method for proving various formulas that describe the shifts in the scattering resonances due to small particles is based on pole-pencil decompositions (see, for instance, [5, 7]) of the volume integral operator associated with the radiating dielectric cavity problem.

The new technique introduced in this paper cannot be easily extended to the transverse magnetic case considered in [1] due to the hyper-singular character of the associated volume-integral operator.

The paper is organized as follows. In Sect. 2, we characterize the scattering resonances of dielectric cavities in terms of the spectrum of a volume integral operator. In Sect. 3, using the method of pole-pencil decompositions, we derive the leading-order term in the shifts of scattering resonances of an open dielectric cavity due to the presence of internal particles. In Sect. 4, using a Lippmann–Schwinger representation formula for the Green’s function associated with the open cavity, we generalize the formula obtained in Sect. 3 to the case of external particles. In Sect. 5, we consider the perturbation of an open dielectric cavity by subwavelength resonant particles. The formula obtained for the shifting of the frequencies shows a strong enhancement in the frequency shift in the case of subwavelength resonant particles. In Sect. 6, we perform an asymptotic analysis for the shift of exceptional scattering resonances. The paper ends with some concluding remarks.

2 Scattering resonances of a dielectric cavity

2.1 Model

We consider the scattering of linearly polarized light by a dielectric cavity in a time-harmonic regime. Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\) for \(d=2,3,\) with smooth boundary \(\partial \Omega \). Assume \(\varepsilon \equiv \tau \varepsilon _c + \varepsilon _m\) inside \(\Omega \) and \(\varepsilon = \varepsilon _m\) outside \(\Omega \), and \(\mu = \mu _m\) everywhere. Here, \(\varepsilon _c,\varepsilon _m,\) and \(\tau \) are positive constants. Since we are interested in scattering resonances, we look for solutions u of the homogeneous Helmholtz equation at complex frequency \(\omega \):

$$\begin{aligned} \left\{ \begin{aligned}&\Delta u + \omega ^2 \varepsilon (x) \mu _m u =0 \qquad \text {in} \ {\mathbb {R}}^d ,\\&u \text { satisfies the outgoing radiation condition.} \end{aligned}\right. \end{aligned}$$
(1)

It is known that the above scattering problem attains a unique solution for \(\omega \) with \(\mathfrak {I}\omega \ge 0\). Using analytic continuation, the solution also exists and is unique for all complex \(\omega \) except for a countable number of points, which are the scattering resonances (see, for instance, [19]).

Let \(\Gamma _m\) be the outgoing fundamental solution of \(\Delta + \varepsilon _m \mu _m \omega ^2\) in free space. We define the following integral operator:

Definition 2.1

Let

$$\begin{aligned} L^2(\Omega ) \longrightarrow&L^2(\Omega ) \\ u \longmapsto&K_\Omega ^\omega [u]:=- \mathop {\int }\limits _\Omega u(y) \Gamma _m (\, \cdot \, -y;\omega ) \mathrm{d}y. \end{aligned}$$

The following Lippmann–Schwinger representation formula holds:

Proposition 2.2

u is a solution of (1) if and only if the restriction of u on \(\Omega \) is a solution of

$$\begin{aligned} \left( I - \omega ^2 \tau \varepsilon _c \mu _m K_\Omega ^\omega \right) [u] = 0, \end{aligned}$$
(2)

where I is the identity operator.

According to [12], the following spectral decomposition of the operator \(K_\Omega ^\omega \) holds:

Lemma 2.3

For \(\omega \in {\mathbb {C}}\), the operator \(K_\Omega ^\omega \) is bounded from \(L^2(\Omega )\) into \(H^2(\Omega )\). Moreover, it is a Hilbert–Schmidt operator. Therefore, its spectrum is

$$\begin{aligned} \sigma (K_\Omega ^\omega ) = \left\{ 0, \lambda _1(\omega ), \lambda _2(\omega ), \ldots , \lambda _j(\omega ), \ldots \right\} , \end{aligned}$$

where \(|\lambda _j(\omega )| \rightarrow 0\) as \(j\rightarrow +\infty \) and \(\{0\} = \sigma (K_\Omega ^\omega ) \setminus \sigma _p(K_\Omega ^\omega )\) with \(\sigma _p(K_\Omega ^\omega )\) being the point spectrum.

Let \(H_j\) be the generalized eigenspace associated with \(\lambda _j(\omega )\). Then, again from [12], it follows that \(L^2(\Omega ) \) is the closure of \(\bigcup _j H_j\).

Lemma 2.4

We have

$$\begin{aligned} L^2(\Omega ) = \overline{\bigcup _j H_j}. \end{aligned}$$

Moreover, if we assume that for any j, \(\text {dim }H_j=1\), and denote by \(e_j\) a unitary basis vector for \(H_j\), then the functions

$$\begin{aligned} f_{j,k}(x,y) = e_j(x)e_k(y), \end{aligned}$$

form a normal basis for \(L^2(\Omega \times \Omega )\) and the following completeness relation holds:

$$\begin{aligned} \delta (x-y) = \sum _j e_j(x) e_j(y). \end{aligned}$$

Remark 2.5

Note that \(\mathfrak {I}\lambda _j(\omega ) \ne 0\) for all j and \(\omega \in {\mathbb {R}}\) because of the Rellich lemma.

Since \(K_\Omega ^\omega \) is a holomorphic family of compact operators for \(\omega \in {\mathbb {C}}\) and \(\left( I - \omega ^2 \tau \varepsilon _c \mu _m K_\Omega ^\omega \right) ^{-1}\) exists for \(\omega \in {\mathbb {R}}\), then by the Fredholm analytic theory, \(\left( I - \omega ^2 \tau \varepsilon _c \mu _m K_\Omega ^\omega \right) ^{-1}\) is a meromorphic family of operators for \(\omega \in {\mathbb {C}}\).

Definition 2.6

In view of Lemmas 2.3 and 2.4, we say that \(\omega _0\) is a scattering resonance for the open cavity problem if there exists a \(j_0\) such that

$$\begin{aligned} 1 - \omega _0^2 \tau \varepsilon _c \mu _m \lambda _{j_0} (\omega _0) = 0. \end{aligned}$$
(3)

We say that the scattering resonance \(\omega _0\) is a non-exceptional scattering resonance if the following assumptions hold:

  1. (i)

    We have

    $$\begin{aligned} 1 - \omega ^2 \tau \varepsilon _c \mu _m \lambda _{j_0} (\omega ) = R(\omega ) (\omega - \omega _0), \end{aligned}$$

    where \(R(\omega _0) \ne 0\) and \(\omega \mapsto R(\omega )\) is holomorphic;

  2. (ii)

    The generalized eigenspace \(H_{j_0}(\omega )\) is of dimension 1.

Remark 2.7

Note that the assumption \(\varepsilon > \varepsilon _m\) in \(\Omega \) is to insure that the imaginary parts of the scattering resonances converge to zero as \(\tau \) goes to infinity (see, for instance, [23]) and therefore, shifts due to the presence of small particles are measurable.

2.2 Pole pencil decomposition of the Green’s function

We denote by \( G(x,y; \omega ) \) the Green’s function associated with problem (1), that is, the solution in the sense of distributions of

$$\begin{aligned} \left( \Delta _x + \omega ^2 \varepsilon (x) \mu _m \right) G(x,y,\omega ) = \delta _y , \end{aligned}$$

satisfying the outgoing radiation condition.

We can give the following expansion for G when \(\omega \) is close to a non-exceptional scattering resonance. We refer to “Appendix A” for its proof.

Proposition 2.8

Assume that \(\omega _0\) is a non-exceptional scattering resonance. There exists a complex neighborhood \(V(\omega _0)\) of \(\omega _0\) such that for \(\omega \) in \(V(\omega _0) \setminus \{\omega _0\}\),

$$\begin{aligned} G(x,y; \omega ) = \Gamma _m(x-y;\omega ) + c_{j_0}(\omega _0) \dfrac{e_{j_0}(x;\omega ) e_{j_0} (y;\omega )}{\omega - \omega _0} + {\widetilde{R}}(x,y;\omega ), \end{aligned}$$
(4)

where \(\mathrm {vect}(e_{j_0})= H_{j_0}\). Moreover, \(\omega \mapsto {\widetilde{R}}(x,y;\omega )\), \(\omega \mapsto e_{j_0}(\,\cdot , \omega )\), and \(\omega \mapsto c_{j_0}(\omega )\) are all holomorphic in \(V(\omega _0)\), and \((x,y) \mapsto {\widetilde{R}}(x,y;\omega )\) is smooth.

3 Shift of the scattering resonances by internal small particles

Now let \(D \Subset \Omega \) be a small particle of the form \(D= z+\delta B\), where \(\delta \) is the characteristic size of D, z is its location, and B is a smooth bounded domain containing the origin. We suppose that D has a material parameter \(\mu _c\) that is different from \(\mu _m\), and consider the operator

$$\begin{aligned} \nabla \cdot \dfrac{1}{\mu } \nabla + \varepsilon \omega ^2, \end{aligned}$$

where \(\mu = \mu _c\) in D and \(\mu = \mu _m\) outside D.

As \(\delta \rightarrow 0\), we seek an \(\omega _\delta \) in a neighborhood of \(\omega _0\) such that there exists a non-trivial solution to

$$\begin{aligned} (\nabla \cdot \dfrac{1}{\mu } \nabla + \varepsilon \omega ^2_\delta ) u = 0, \end{aligned}$$
(5)

subject to the outgoing radiation condition.

The following asymptotic expansion of \(\omega _\delta \) holds.

Proposition 3.1

Assume that \(\omega _0\) is a non-exceptional scattering resonance. Then, as \(\delta \rightarrow 0\), we have

$$\begin{aligned} \omega _\delta - \omega _0 \simeq \delta ^d c_{j_0}(\omega _0) M(\mu _m / \mu _c,B) \nabla e_{j_0}(z;\omega _0) \cdot \nabla {e}_{j_0}(z; \omega _0), \end{aligned}$$
(6)

where M is the polarization tensor given by

$$\begin{aligned} M(\mu _m / \mu _c,B) = (\frac{\mu _m}{\mu _c} -1) \mathop {\int }\limits _{\partial B} \frac{\partial v^{(1)}}{\partial \nu } \big |_{-} (\xi ) \xi \, \mathrm{d}\sigma (\xi ), \end{aligned}$$
(7)

with \(v^{(1)}\) being such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _\xi v^{(1)} = 0 &{} \text{ in } {\mathbb {R}}^d \setminus {\bar{B}},\\ \Delta _\xi v^{(1)} = 0 &{} \text{ in } B, \\ {v^{(1)} |_+ = v^{(1)} |_{-}} &{} {\text{ on } \partial B,}\\ \dfrac{\partial v^{(1)}}{\partial \nu }|_+ = (\mu _m/\mu _c) \dfrac{\partial v^{(1)}}{\partial \nu } |_- &{} \text{ on } \partial B,\\ v^{(1)}(\xi ) \sim \xi &{} \text{ as } |\xi | \rightarrow +\infty . \end{array}\right. } \end{aligned}$$
(8)

Before proving the above result, we state the following useful lemma. We refer to “Appendix B” for its proof.

Lemma 3.2

Let

$$\begin{aligned} T_D^\omega : v \mapsto \nabla _x \mathop {\int }\limits _D v(y) \cdot \nabla G(x-y; \omega ) \mathrm{d}y . \end{aligned}$$

Then, \(T_D^{{\omega }}\) is a well-defined operator from \(L^2(D)\) into itself.

Proof (of Proposition 3.1)

The outgoing solution to problem (5) admits the following Lippmann–Schwinger representation formula:

$$\begin{aligned} u(x) = (\frac{\mu _m}{\mu _c} - 1) \mathop {\int }\limits _D \nabla u(y) \cdot \nabla G(x,y;\omega _\delta ) \mathrm{d}y \quad \text{ for } \text{ all } x \in {\mathbb {R}}^d. \end{aligned}$$
(9)

\(\square \)

From, for instance [9, Appendix B], the operator \(T_D^\omega \) is well defined. Therefore, we seek \(\omega _\delta \) such that there is a non-trivial \(v \in L^2(D)^d\) satisfying

$$\begin{aligned} v(x) - (1/\mu _c - 1/\mu _m) T_D^{\omega _\delta }[v] (x) = 0 \quad \text { for all } x \in D, \end{aligned}$$

or equivalently,

$$\begin{aligned} \big (I - (\frac{\mu _m}{\mu _c} - 1) T_D^{\omega _\delta }\big ) [v]=0 \end{aligned}$$
(10)

Hence, as the characteristic size \(\delta \) of D goes to zero, we seek \(\omega _\delta \) in a neighborhood of \(\omega _0\) such that \(1/( (\mu _m/\mu _c) - 1)\) is an eigenvalue of \(T_D^{\omega _\delta }\).

From the pole-pencil decomposition (4) of G, we have

$$\begin{aligned} \nabla \mathop {\int }\limits _D v \cdot \nabla G = \nabla \mathop {\int }\limits _D v \cdot \nabla \Gamma _m + \dfrac{c_{j_0} (\omega )}{\omega - \omega _0} \big (\mathop {\int }\limits _D v \cdot \nabla {e}_{j_0} \, \mathrm{d}y \big ) \nabla {e_{j_0}} (x;\omega ) + R[v], \end{aligned}$$

where \(R: L^2(D)^d \rightarrow L^2(D)^d\) is an operator with smooth kernel that is holomorphic in \(\omega \in V(\omega _0).\) Let

$$\begin{aligned} N_D^\omega : v \in L^2(D)^d \mapsto \nabla _x \mathop {\int }\limits _D v(y) \cdot \nabla \Gamma _m(x-y;\omega ) \mathrm{d}y \in L^2(D)^d. \end{aligned}$$

Then, it follows that

$$\begin{aligned} \begin{array}{lll} \displaystyle \frac{1}{\frac{\mu _m}{\mu _c} - 1} \bigg (I - (\frac{\mu _m}{\mu _c} - 1) T_D^\omega \bigg )[v] &{} = &{} \displaystyle \bigg ( \frac{I}{\frac{\mu _m}{\mu _c} - 1} - N_D^\omega \bigg )[v] \\ &{}&{} \displaystyle -\, \dfrac{c_{j_0}(\omega )}{\omega - \omega _0} (v, \nabla {e}_{j_0}) \nabla e_{j_0} {-} R[v], \end{array} \end{aligned}$$

where \((\, \cdot , \cdot \,)\) denotes the \(L^2\) real scalar product on D.

Let

$$\begin{aligned} L = 1/( (\mu _m/\mu _c) - 1) I - N_D^{0}, \end{aligned}$$
(11)

where \(N_D^0:= N_D^{\omega =0}\). Then, (10) can be rewritten as:

$$\begin{aligned} L[v] - \dfrac{c_{j_0}(\omega )}{\omega - \omega _0} (v, \nabla {e}_{j_0}) \nabla e_{j_0} + {\widetilde{R}}[v] = 0 , \end{aligned}$$

where \({\widetilde{R}}: L^2(D)^d \rightarrow L^2(D)^d\) is an operator with smooth kernel that is holomorphic in \(\omega \in V(\omega _0).\)

Now, we make use of the orthogonal decomposition of \(L^2(D)\) and the spectral analysis of \(N_D^0\) on \(L^2(D)\) that can be found in [17, 18]. More precisely, recall that

$$\begin{aligned} L^2\left( D\right) = \nabla H^1_0(D) \oplus {H}(\mathrm {div\ }0,D) \oplus {W}, \end{aligned}$$

where \( H^1_0(D)\) is the set of \(H^1\)-functions in D with trace zero on \(\partial D\), \({H}(\mathrm {div\ }0,D)\) is the space of divergence free \(L^2\)-vector fields and W is the space of gradients of harmonic \(H^1\) functions. Here, \(H^1\) is the set of functions in \(L^2\) having their weak derivatives in \(L^2\). We will use the following lemma proved in [9]:

Lemma 3.3

The operator \(N_D^0\) is a bounded self-adjoint map on \(L^2(D)\) with \( \nabla H^1_0(D)\), \({H}(\text {div } 0,D)\) and W as invariant subspaces. On \(\nabla H^1_0(\Omega )\), \(N_D^0[\phi ]=\phi \), on \({H}(\text {div } 0,D)\), \(N_D^0[\phi ]=0\) and on W:

$$\begin{aligned} \nu \cdot N_D^0[\phi ]= \left( \frac{1}{2} + {\mathcal {K}}_D^*\right) [\phi \cdot \nu ] \mathrm {\ on\ } \partial D , \end{aligned}$$

where \(\nu \) is the outward normal on \(\partial D\) and \(K_D^*: L^2(\partial D) \rightarrow L^2(\partial D)\) is the Neumann–Poincaré operator associated with \(\partial D\). Recall that \(K_D^*\) is given for \(\varphi \in L^2(\partial D)\) by

$$\begin{aligned} K_D^*[\varphi ]= \mathop {\int }\limits _{\partial D} \frac{\partial \Gamma ^{(0)}(x,y)}{\partial \nu (x)} \varphi (y)\, \mathrm{d}\sigma (y), \end{aligned}$$

where \(\Gamma ^{(0)}\) is the fundamental solution of the Laplacian in \({\mathbb {R}}^d\).

Moreover, \((1/2) I - N_D^0|_W: {W} \longrightarrow {W}\) is a compact operator and hence, its spectrum is discrete and the associated eigenfunctions form a basis of W.

We refer the reader to [5] for the properties of the Neumann–Poincaré operator \(K_D^*\).

Therefore, using Lemma 3.3, we have

$$\begin{aligned} v - \dfrac{c_{j_0}(\omega )}{\omega - \omega _0} (v, \nabla {e}_{j_0}) L^{-1}[\nabla e_{j_0}] + L^{-1} {\widetilde{R}} [v] = 0. \end{aligned}$$

So, since

$$\begin{aligned} || L^{-1} {\widetilde{R}}||_{{\mathcal {L}}(L^2(D)^d, L^2(D)^d)} =o(1) \quad \text{ as } \delta \rightarrow 0, \end{aligned}$$

see [7] and [9, Lemma 4.2], the term \(L^{-1} {\widetilde{R}}[v]\) can be neglected, and the following asymptotic expansion holds:

$$\begin{aligned} \omega _\delta - \omega _0 \simeq c_{j_0}(\omega _0) (L^{-1}[\nabla e_{j_0}], \nabla {e}_{j_0}). \end{aligned}$$

Moreover, from [9, Proposition 3.1] (see also “Appendix C”), it follows that

$$\begin{aligned} (L^{-1}[\nabla e_{j_0}], \nabla {e}_{j_0}) \simeq \delta ^d M(\mu _m / \mu _c,B) \nabla e_{j_0}(z; \omega _0) \cdot \nabla {e}_{j_0}(z; \omega _0), \end{aligned}$$
(12)

where M is the polarization tensor given by (7); see [6]. The proof is then complete. \(\square \)

To conclude this section, it is worth mentioning that in the case where the parameter \(\varepsilon \) inside the small particle is different from the background one, an asymptotic formula for the shift of the scattering resonance can be derived. Say, for instance, that the parameter inside the particle, which we denote by \(\varepsilon _D\), is different from the background parameter. Then, by extending the representation formula (9) to this case, we can show that \(\omega _\delta -\omega _0\) can be approximated by

$$\begin{aligned} \omega _\delta - \omega _0 \simeq \delta ^d c_{j_0}(\omega _0) M(\mu _m / \mu _c,B) \nabla e_{j_0}(z;\omega _0) \cdot \nabla {e}_{j_0}(z; \omega _0) + c_{j_0}(\omega _0) |D| \omega _0^2 (\varepsilon _D - \varepsilon ) (e_{j_0}(z;\omega _0))^2. \end{aligned}$$

4 Shift of the scattering resonances by external small particles

Now consider the case where the particle is outside \(\Omega \). The main difference in this case is that the modes of \(K_\Omega ^\omega \) are not defined on D, and therefore, we must first write the expansion for G outside of \(\Omega \). We start by recalling the Lippmann–Schwinger equation for \(v=G-\Gamma _m\):

$$\begin{aligned} \left( I-\omega ^2\tau \varepsilon _c \mu _m K_\Omega ^\omega \right) [v(\cdot ,x_0)](x)=\omega ^2\tau \varepsilon _c \mu _m K_\Omega ^\omega \left[ \Gamma _m(\cdot ,x_0) \right] (x) \qquad \text{ for } x,x_0\in \Omega . \end{aligned}$$
(13)

Now, using Proposition 2.8 for z and \(z'\) inside \(\Omega \), we have

$$\begin{aligned} v(z,z';\omega )= c_{j_0}(\omega ) \dfrac{e_{j_0}(z;\omega ) e_{j_0}(z';\omega )}{\omega -\omega _0}+ {\widetilde{R}}(z,z';\omega ), \end{aligned}$$

and we can write an expansion for \(v(x,x_0)\) for \(x\in {\mathbb {R}}^d \setminus \Omega \):

$$\begin{aligned}&v(x,x_0) -\frac{ \omega ^2\tau \varepsilon _c \mu _m c_{j_0}(\omega )}{\omega -\omega _0}\mathop {\int }\limits _\Omega e_{j_0}(z,\omega )\Gamma _m(z,x)e_{j_0}(x_0,\omega ) \mathrm{d}z - \omega ^2\tau \varepsilon _c \mu _m K_\Omega ^\omega [{\widetilde{R}}(\cdot ,x_0;\omega )](x)\\&\quad = \omega ^2\tau \varepsilon _c \mu _m K_\Omega ^\omega \left[ \Gamma _m(\cdot ,x_0) \right] (x) \end{aligned}$$

for \(x \in {\mathbb {R}}^d ,x_0\in \Omega .\) The latter equality can be written as:

$$\begin{aligned} v(x,x_0) = \frac{ \omega ^2\tau \varepsilon _c \mu _m c_{j_0}(\omega )}{\omega -\omega _0}\left( \mathop {\int }\limits _\Omega e_{j_0}(z,\omega )\Gamma _m(z,x) \mathrm{d}z\right) e_{j_0}(x_0,\omega ) + R_1(x,x_0,\omega ) \end{aligned}$$

for \(x \in {\mathbb {R}}^d ,x_0\in \Omega \), where \(R_1\) is regular in space and holomorphic in \(\omega \). Let

$$\begin{aligned} g_{j_0}(x;\omega ) := \omega ^2 \tau \varepsilon _c \mu _m \mathop {\int }\limits _\Omega e_{j_0} (z';\omega ) \Gamma _m(z',x;\omega ) \mathrm{d}z',\qquad x\in {\mathbb {R}}^d . \end{aligned}$$
(14)

We have

$$\begin{aligned} v(x,x_0) = \frac{ c_{j_0}(\omega )}{\omega -\omega _0}g_{j_0}(x;\omega ) e_{j_0}(x_0,\omega ) + R_1(x,x_0,\omega ), \qquad x \in {\mathbb {R}}^d ,x_0\in \Omega . \end{aligned}$$
(15)

Now, let \( x,x_0\in {\mathbb {R}}^d\). By using the Lippmann–Schwinger equation (13), it follows that

$$\begin{aligned} \begin{array}{lll} \displaystyle v(x,x_0) &{} = &{} \displaystyle { \omega ^2\tau \varepsilon _c \mu _m} \mathop {\int }\limits _\Omega \Gamma _m(x,z) v(x_0,z) \, dz \\ &{}&{} +\, \omega ^2\tau \varepsilon _c \mu _m \mathop {\int }\limits _\Omega \Gamma _m(x,z) \Gamma _m(x_0,z) \, dz. \end{array} \end{aligned}$$

We can now use expansion (15) to obtain that

$$\begin{aligned}&v(x,x_0) - \frac{ \omega ^2\tau \varepsilon _c \mu _m c_{j_0}(\omega )}{\omega -\omega _0} g_{j_0}(x_0;\omega ) \left( \mathop {\int }\limits _\Omega e_{j_0}(z,\omega )\Gamma _m(x,z) \mathrm{d}z\right) - \omega ^2\tau \varepsilon _c \mu _m K_\Omega ^\omega [R_1(\cdot ,x,\omega )](x_0) \\&\quad = \omega ^2\tau \varepsilon _c \mu _m K_\Omega ^\omega \left[ \Gamma _m(\cdot ,x) \right] (x_0), \qquad x \in {\mathbb {R}}^d ,x_0\in {\mathbb {R}}^d. \end{aligned}$$

Therefore, we have an expansion for v outside of \(\Omega \):

$$\begin{aligned} v(x,x_0) = \frac{ c_{j_0}(\omega )}{\omega -\omega _0}g_{j_0}(x;\omega ) g_{j_0}(x_0;\omega ) + R_2(x,x_0,\omega ), \qquad x \in {\mathbb {R}}^d ,x_0\in {\mathbb {R}}^d. \end{aligned}$$

Analogous to the calculations in the previous section, we have

$$\begin{aligned} v - \dfrac{c_{j_0} (\omega )}{\omega - \omega _0} ( v, \nabla g_{j_0}) L^{-1}[\nabla g_{j_0}] + L^{-1} R[v] = 0, \end{aligned}$$

for some operator R with smooth kernel that is holomorphic in \(\omega \) in a neighborhood \(V(\omega _0)\) of \(\omega _0\). Therefore, by exactly the same method as in the previous section, the following asymptotic expansion can be obtained.

Proposition 4.1

Assume that \(\omega _0\) is a non-exceptional scattering resonance. Then, as \(\delta \rightarrow 0\), we have

$$\begin{aligned} \omega _\delta - \omega _0 \simeq \delta ^d c_{j_0}(\omega _0) M(\mu _m /\mu _c,B) \nabla g_{j_0}(z;\omega _0) \cdot \nabla {g}_{j_0} (z;\omega _0), \end{aligned}$$
(16)

where \({g}_{j_0}\) is defined by (14) and \( M(\mu _m /\mu _c,B)\) is given by (7).

5 Shift of the scattering resonances due to resonant particles

Let \(D \Subset \Omega \). Suppose that D is such that \(\mu _c\) depends on \(\omega \), and, for a discrete set of frequencies \(\omega \), problem (8) (or equivalently the operator \(\displaystyle {\big (\frac{\mu _m+\mu _c(\omega )}{2(\mu _m - \mu _c(\omega ))} I- K_D^* \big )}\)) is singular, see [4, 10, 11]. We call such frequencies subwavelength resonances. In that case, we have the following scattering resonance problem: Find \(\omega \) such that there is a non-trivial solution v to

$$\begin{aligned} L(\omega ) [v] - \dfrac{c_{j_0}(\omega )}{\omega -\omega _0} ( v, \nabla e_{j_0}) \nabla e_{j_0} + R[v] = 0 , \end{aligned}$$
(17)

where \(L(\omega )\) is defined by (11). Using, for instance, the Drude model for \(\mu _c\), we have \(\mu _c(\omega ) = \mu _m (1 - \omega _p^2 / \omega ^2)\), where \(\omega _p\) is a given real constant.

It is easy to see that the singular character of (8) is linked to the non-invertibility of \(L(\omega )\) on W.

Denote by \(P_1\ :\ L^2(D) \longrightarrow L^2(D) \) the orthogonal projector on \(\nabla H^1_0(D)\) and \(P_2\ : \ L^2(D) \longrightarrow L^2(D) \) the orthogonal projector on \({H}(\mathrm {div\ }0,D)\). Using Lemma 3.3, we can write the resolvent operator \(L^{-1}(\omega )\) as follows:

$$\begin{aligned} L(\omega )^{-1} = \frac{1}{1-\lambda (\omega )} P_1 + \frac{1}{\lambda (\omega )}P_2+\sum _j \dfrac{( \cdot , \varphi _j) \varphi _j}{\lambda (\omega ) - \lambda _j}, \end{aligned}$$

where \((\lambda _j, \varphi _j)_j\) are the pairs of eigenvalues and associated orthonormal eigenfunctions of \(N_D^0\). We can then rewrite equation (17) as follows:

$$\begin{aligned} v - \dfrac{c_{j_0}(\omega )}{\omega - \omega _0} \dfrac{( v , \nabla e_{j_0}) ( \nabla e_{j_0}, \varphi _j) \varphi _j}{\lambda (\omega ) - \lambda _j} + L^{-1}R[v]= 0. \end{aligned}$$

Now, taking the scalar product on \(L^2(D)\) with \(\nabla e_{j_0}\) and multiplying by \((\omega -\omega _0) (\lambda (\omega )-\lambda _j)\), we obtain that

$$\begin{aligned} (\omega -\omega _0) (\lambda (\omega )-\lambda _j) ( v,\nabla e_{j_0}) - c_{j_0} (\omega _0) (v,\nabla e_{j_0}) (\nabla e_{j_0}, \varphi _j)^2 + (\omega -\omega _0) (\lambda (\omega )-\lambda _j) L^{-1}R[v] = 0. \end{aligned}$$

Since R is holomorphic in \(\omega \), the remainder \((\omega -\omega _0) (\lambda (\omega )-\lambda _j) L^{-1}R[v]\) is negligible in a neighborhood of \(\omega _0\). Hence, we arrive at the following proposition:

Proposition 5.1

As \(\delta \rightarrow 0\), we have

$$\begin{aligned} (\omega _\delta - \omega _0) (\lambda (\omega _\delta ) - \lambda _j) \simeq c_{j_0}(\omega _0) (\nabla e_{j_0}, \varphi _j)^2. \end{aligned}$$

Note that if \(\lambda (\omega ) - \lambda _j \simeq O(\omega - \omega _0)\) for \(\omega \) close to \(\omega _0\), then we obtain

$$\begin{aligned} (\omega _\delta - \omega _0)^2 \simeq c_{j_0}(\omega _0) ( \nabla e_{j_0}(\cdot ; \omega _0), \varphi _j)^2. \end{aligned}$$

Hence, we have a significant shift in the scattering resonances if the particle D is resonant near or at the frequency \(\omega _0\). In fact, the shift in the scattering resonance is proportional to the square root of the particle’s volume. This anomalous effect has been observed in [24].

6 Asymptotic analysis near exceptional scattering resonances

In this section, we consider the asymptotic behavior of an exceptional scattering resonance for a particular form of the Green’s function. These exceptional resonances are due to the non-Hermitian character of the operator \(T_D^\omega \), see [12, 22]. For simplicity and in view of the Jordan-type decomposition of the operator \(T_D^\omega \) established in [12], we assume that, for \(\omega \) near \(\omega _0\), \(G(x,y;\omega )\) behaves like

$$\begin{aligned} G(x,y;\omega ) = \Gamma _m(x,y;\omega ) + c_1(\omega ) \dfrac{ h^{(1)}(x;\omega ) h^{(1)}(y;\omega )}{\omega - \omega _0} + c_2(\omega ) \dfrac{ h^{(2)} (x;\omega ) h^{(2)}(y;\omega )}{(\omega - \omega _0)^2} + R(x,y;\omega ), \end{aligned}$$
(18)

for two functions \(h^{(1)}\) and \(h^{(2)}\) in \(L^2(D)\). Here, the functions \(\omega \mapsto c_j(\omega )\), \(j=1,2\) and \(\omega \mapsto R(x,y;\omega ) \) are all holomorphic in a neighborhood of \(\omega _0\), and \((x,y) \mapsto {R}(x,y; \omega )\) is smooth.

In this simple case, where the exceptional scattering resonance is of second order, we characterize the splitting of the scattering resonance \(\omega _0\) due to the small particle D, which is assumed for simplicity to be non-resonant.

Following the same arguments as those in the previous sections, we neglect R in (18) and seek a non-trivial v such that

$$\begin{aligned} L[v] - c_1(\omega ) \dfrac{(v, \nabla h^{(1)}) }{\omega - \omega _0}\nabla {h^{(1)}} - c_2(\omega ) \dfrac{(v, \nabla h^{(2)}) }{(\omega - \omega _0)^2} \nabla {h^{(2)}} = 0 , \end{aligned}$$

or equivalently,

$$\begin{aligned} v - c_1(\omega ) \dfrac{(v, \nabla h^{(1)}) }{\omega - \omega _0} L^{-1} [\nabla {h^{(1)}}] - c_2(\omega ) \dfrac{(v, \nabla h^{(2)})}{(\omega - \omega _0)^2} L^{-1}[\nabla {h^{(2)}}] = 0 . \end{aligned}$$

By multiplying the above equations by \(\nabla h^{(1)}\) and \(\nabla h^{(2)}\), respectively, and integrating by parts over D, we obtain the following system of equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} (v, \nabla h ^{(1)}) \Bigg ( 1 - c_1(\omega ) \dfrac{(L^{-1}[\nabla {h^{(1)}}] , \nabla h^{(1)}) }{\omega - \omega _0 } \Bigg )= c_2(\omega ) (v, \nabla h^{(2)}) \dfrac{(L^{-1}[\nabla {h^{(2)}}], \nabla h ^{(1)})}{(\omega - \omega _0)^2},\\ (v, \nabla h^{(2)}) \Bigg (1 - c_2(\omega ) \dfrac{(L^{-1}[\nabla {h^{(2)}}], \nabla h^{(2)})}{(\omega - \omega _0)^2} \Bigg ) = c_1(\omega ) (v, \nabla h ^{(1)})\dfrac{ (L^{-1}[\nabla {h^{(1)}}], \nabla h^{(2)})}{\omega - \omega _0}. \end{array}\right. } \end{aligned}$$

Therefore, the following result holds.

Proposition 6.1

Assume that the decomposition (18) holds for \(\omega \) near \(\omega _0\). Then, the perturbed scattering resonance problem (due to the particle D) can be approximately reformulated as a search for \(\omega \) near \(\omega _0\) such that the matrix

$$\begin{aligned} {\mathcal {A}}(\omega ):= \begin{pmatrix} 1 - c_1(\omega ) \dfrac{(L^{-1}[\nabla {h^{(1)}}] , \nabla h^{(1)}) }{\omega - \omega _0 } &{} -c_2(\omega )\dfrac{(L^{-1}[\nabla {h^{(2)}}], \nabla h ^{(1)})}{(\omega - \omega _0)^2} \\ c_1(\omega ) \dfrac{ (L^{-1}[\nabla {h^{(1)}}], \nabla h^{(2)})}{\omega - \omega _0} &{} 1 - c_2(\omega ) \dfrac{(L^{-1}[\nabla {h^{(2)}}], \nabla h^{(2)})}{(\omega - \omega _0)^2} \end{pmatrix} \end{aligned}$$

is singular.

In view of Proposition 6.1, the second-order exceptional scattering frequency is split into two scattering frequencies which can be computed approximately by finding the values of \(\omega \) for which the determinant of the matrix \({\mathcal {A}}(\omega )\) is zero. When D is a disk or a sphere, the functions \(h^{(j)}\) and the functions \(c_j(\omega )\) for \(j=1,2\), defined in (18), can be computed explicitly for \(\omega \) near a resonance \(\omega _0\) (see, for instance, [8]) and hence, an expression for \({\mathcal {A}}(\omega )\) can be obtained. In the general case, it seems difficult to obtain accurate approximations of the functions \(h^{(j)}\) and the functions \(c_j(\omega )\) for \(j=1,2\).

Assume that \((L^{-1}[\nabla {h^{(1)}}], \nabla h^{(2)}) =0.\) Then, the values of \(\omega \) near \(\omega _0\) such that the determinant of \({\mathcal {A}}(\omega )\) is zero are determined by

$$\begin{aligned} (\omega - \omega _0)^2 = c_j(\omega ) (L^{-1}[\nabla {h^{(j)}}], \nabla h^{(j)}), \end{aligned}$$

for \(j=1\) or 2. Since \(c_j(\omega ) (L^{-1}[\nabla {h^{(j)}}], \nabla h^{(j)}) = O(|D|)\) for \(j=1,2,\) and \(\omega \) near \(\omega _0\), it can be easily seen that the splitting corresponding to \(j=2\) of \(\omega _0\) is of order the square root of the volume of the particle. This is in contrast with (6), where the perturbation induced by the small particle is proportional to its volume.

It is worth emphasizing that the derivations presented in this section can be generalized to the case of exceptional points of arbitrary order N. In this case, it is expected that the strength of the splitting induced by a small particle on an exceptional scattering frequency of order N is proportional to the volume of the particle to the power 1/N.

7 Concluding remarks

In this paper, the leading-order term in the shifts of scattering resonances of a radiating dielectric cavity due to the presence of small particles is derived. The formula describes the dependency of the frequency shifts on the position and the polarization tensor of the particle. It is also proved that the shift is significantly enhanced if the particle is a subwavelength resonant particle which resonates near or at a scattering resonance of the cavity. A characterization of the splitting of the scattering resonances due to small particles near an exceptional scattering resonance is performed. It would be challenging to develop a general theory near such frequencies. This would be the subject of a forthcoming paper.