Pavel Exner, Jiří Lipovský
We discuss resonances for Schrödinger operators on metric graphs which consists of a finite compact part and a finite number of halflines attached to it; the vertex coupling is assumed to be of the $δ$-type or certain modifications of it. Using exterior complex scaling on the graph we show that the resolvent and scattering resonances coincide in this case.
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We establish semiclassical resolvent estimates for Schrödinger operators with long-range matrix-valued potentials. As an application we prove resonance free domains both in trapping and non-trapping situations. Our results generalize the well-known results of \cite{Burq1, Martinez1} in the case of scalar Schrödinger operators.
In this note, we consider semiclassical scattering on a manifold which is Euclidean near infinity or asymptotically hyperbolic. We show that, if the cut-off resolvent satisfies polynomial estimates in a strip of size $O(h |\log h|^{-α})$ below the real axis, for some $α\geq 0$, then the cut-off resolvent is actually bounded by $O(|\log h|^{α+1} h^{-1})$ in this strip. As an application, we improve slightly the estimates on the real axis given by Bourgain and Dyatlov in the case of convex co-compact surfaces.
Using coordinates $(x,y)\in \mathbb R\times \mathbb R^{d-1}$, we introduce the notion that an unbounded domain in $\mathbb R^d$ is star shaped with respect to $x=\pm \infty$. For such domains, we prove estimates on the resolvent of the Dirichlet Laplacian near the continuous spectrum. When the domain has infinite cylindrical ends, this has consequences for wave decay and resonance-free regions. Our results also cover examples beyond the star-shaped case, including scattering by a strictly convex obstacle inside a straight planar waveguide.
Manifolds with infinite cylindrical ends have continuous spectrum of increasing multiplicity as energy grows, and in general embedded resonances (resonances on the real line, embedded in the continuous spectrum) and embedded eigenvalues can accumulate at infinity. However, we prove that if geodesic trapping is sufficiently mild, then the number of embedded resonances and eigenvalues is finite, and moreover the cutoff resolvent is uniformly bounded at high energies. We obtain as a corollary the existence of resonance free regions near the continuous spectrum. We also obtain improved estimates w…
In this paper we extend the results proved in (Carlone, R., Correggi, M., Finco, D., Teta, A.: A model for Feshbach Resonances arXiv:1901.08282 [math-ph]) about Feshbach resonances in a multichannel Hamiltonian $\mathcal{H}$, proving a low energy expansion of the resolvent $(\mathcal{H}-k^{2})^{-1}$ as $k\to 0$ in the resonant case.
We establish high energy $L^2$ estimates for the restriction of the free Green's function to hypersurfaces in $\mathbb{R}^d$. As an application, we estimate the size of a logarithmic resonance free region for scattering by potentials of the form $V\otimes δ_Γ$, where $Γ\subset \mathbb{R}^d$ is a finite union of compact subsets of embedded hypersurfaces. In odd dimensions we prove a resonance expansion for solutions to the wave equation with such a potential.
We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a semiclassical Schrödinger operator is supposed to be smooth and to decrease at infinity, the usual formulas relating scattering quantities and resonances still hold. The main ingredient for the proofs is a resolvent estimate of a new type, relating the resolvent of an operator with the resolvent of its cut-off counterpart.
This paper is devoted to the study of a semiclassical "black box" operator $P$. We estimate the norm of its resolvent truncated near the trapped set by the norm of its resolvent truncated on rings far away from the origin. For $z$ in the unphysical sheet with $- h |ln h| < Im z < 0$, we prove that this estimate holds with a constant $h |Im z|^{-1} e^{C|Im z|/h}$. We also obtain analogous bounds for the resonances states of $P$. These results hold without any assumption on the trapped set neither any assumption on the multiplicity of the resonances.
We consider manifolds with conic singularites that are isometric to $\mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a "very weak" Huygens' principle, which may be of independent interest. As applications of the estimate, we obtain…
Resonance expansions and local-energy decay estimates for Dirac operators
该文研究三维半经典Dirac算子的共振展开与局部能量衰减，通过预解式估计和FBI变换等方法，推导出快速衰减阈值及微局部障碍。
We consider a straight three dimensional quantum layer with singular potential supported on a straight wire which is localized perpendicularly to the walls and connects them. We prove that the infinite number of embedded eigenvalues appears in this system. Furthermore, we show that after introducing a small surface impurity to the layer, the embedded eigenvalues turn to the second sheet resolvent poles which state resonances. We discuss the asymptotics of the imaginary component of the resolvent pole with respect to the surface area.
Resolvent, spectrum and resonances for the acoustic operator with piecewise constant coefficients
本文研究了具有分段常数系数的声学算子的余子式、谱和共振特性，通过极限吸收原理和分析展开式揭示了其谱结构和共振行为。
In this article, we study a class of non-selfadjoint Schr{ö}dinger operators H which are perturbation of some model operator H 0 satisfying a weighted coercive assumption. For the model operator H 0 , we prove that the derivatives of the resolvent satisfy some Gevrey estimates at threshold zero. As application, we establish large time expansions of semigroups e --tH and e --itH for t > 0 with subexponential time-decay estimates on the remainder, including possible presence of zero eigenvalue and real resonances.
We present the Laplace operator associated to a hyperbolic surface $Γ\setminus\mathbb{H}$ and a unitary representation of the fundamental group $Γ$, extending the previous definition for hyperbolic surfaces of finite area to those of infinite area. We show that the resolvent of this operator admits a meromorphic continuation to all of $\mathbb{C}$ by constructing a parametrix for the Laplacian, following the approach by Guillopé and Zworski. We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.
We consider the Schrödinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold $0$. Precise expressions are obtained for the first few coefficients of the expansion in terms of the generalized eigenfunctions. This result justifies the classification of threshold types solely by growth properties of the generalized eigenfunctions. By choosing an appropriate free operator a priori possessing no zero eigenvalue or zero resonance we can simplify th…
We extend the method of multiscale analysis for resonances introduced in [5] in order to infer analytic properties of resonances and eigenvalues (and their eigenprojections) as well as estimates for the localization of the spectrum of dilated Hamiltonians and norm-bounds for the corresponding resolvent operators, in neighborhoods of resonances and eigenvalues. We apply our method to the massless Spin-Boson model assuming a slight infrared regularization. We prove that the resonance and the ground-state eigenvalue (and their eigenprojections) are analytic with respect to the dilation parameter …
In the present paper we investigate the set $Σ_J$ of all $J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices $<2,2>$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein space $(\mathfrak{H}, [\cdot,\cdot])$, SJ=JS. Our aim is to describe different types of $J$-self-adjoint extensions of $S$. One of our main results is the equivalence between the presence of $J$-self-adjoint extensions of $S$ with empty resolvent set and the commutation of $S$ with a Clifford algebra ${\mathcal C}l_2(J,R)$, where $R$ is an additional fundamental symmetry with $JR=-R…
This paper studies the resonances of Schrödinger operators with bounded, compactly supported, real-valued potentials on d-dimensional Euclidean space, where d is even. If the potential V is non-trivial and d is not 4 then the meromorphic continuation of the resolvent of the Schrödinger operator has infinitely many poles, with a quantitative lower bound on their density. A somewhat weaker statement holds if d =4. We prove several inverse-type results. If the meromorphic continuations of the resolvents of two Schrödinger operators $-Δ+V_1$ and $-Δ+V_2$ have the same poles, with both potentials b…
In this paper, we shall show that the limiting absorption principle for the wave operator on the asymptotically Minkowski spacetime. This problem was previously considered by [A. Vasy, J. Spect. Theory, 10,439-461 , (2020)]. Here, we employ a more transparent tool, the Mourre theory and removes an additional condition which is imposed in his paper. Moreover, we also prove that the anti-Feynman propagator defined by Gérard and Wrochna coincides with the outgoing resolvent.
Simplified models of transport in mesoscopic systems are often based on a small sample connected to a finite number of leads. The leads are often modelled using the Laplacian on the discrete half-line $\mathbb N$. Detailed studies of the transport near thresholds require detailed information on the resolvent of the Laplacian on the discrete half-line. This paper presents a complete study of threshold resonance states and resolvent expansions at a threshold for the Schrödinger operator on the discrete half-line $\mathbb N$ with a general boundary condition. A precise description of the expansio…
We study the mathematical theory of quantum resonances in the standard model of non-relativistic QED and in Nelson's model. In particular, we estimate the survival probability of metastable states corresponding to quantum resonances and relate the resonances to poles of an analytic continuation of matrix elements of the resolvent of the quantum Hamiltonian.
We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family $X_\eps$ of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on $X_\eps$ approximate those of the Laplacian with ``free'' boundary conditions on $X_0$, the skeleton graph of $X_\eps$.
We study the large-time behavior of the solutions to the Schrödinger equation associated with a quickly decaying potential in dimension three. We establish the resolvent expansions at threshold zero and near positive resonances. The large-time expansions of solutions are obtained under different conditions, including the existence of positive resonances and zero resonance or/and zero eigenvalue.
We investigate the existence of resonances for two-centers Coulomb systems with arbitrary charges in two dimensions, defining them in terms of generalised complex eigenvalues of a non-selfadjoint deformation of the two-centers Schrödinger operator. We construct the resolvent kernels of the operators and prove that they can be extended analytically to the second Riemann sheet. The resonances are then analysed by means of perturbation theory and numerical methods.
In scattering experiments, physicists observe so-called resonances as peaks at certain energy values in the measured scattering cross sections per solid angle. These peaks are usually associate with certain scattering processes, e.g., emission, absorption, or excitation of certain particles and systems. On the other hand, mathematicians define resonances as poles of an analytic continuation of the resolvent operator through complex dilations. A major challenge is to relate these scattering and resonance theoretical notions, e.g., to prove that the poles of the resolvent operator induce the abo…
We consider the large $L$ limit of one dimensional Schrödinger operators $H_L=-d^2/dx^2 + V_1(x) + V_{2,L}(x)$ in two cases: when $V_{2,L}(x)=V_2(x-L)$ and when $V_{2,L}(x)=e^{-cL}δ(x-L)$. This is motivated by some recent work of Herbst and Mavi where $V_{2,L}$ is replaced by a Dirichlet boundary condition at $L$. The Hamiltonian $H_L$ converges to $H = -d^2/dx^2 + V_1(x)$ as $L\to \infty$ in the strong resolvent sense (and even in the norm resolvent sense for our second case). However, most of the resonances of $H_L$ do not converge to those of $H$. Instead, they crowd together and converge o…
We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asym…
Resonance expansion for quantum walks and its applications to the long-time behavior
本文为整数集上的量子游走引入共振概念，通过共振展开刻画其长时间行为，包括衰减速率、渐近概率分布与弱极限，并探讨共振的通用简单性。
This paper is an electronic application to my set of lectures, subject:`Formal methods in solving differential equations and constructing models of physical phenomena'. Addressed, mainly: postgraduates and related readers. Content: a very detailed discussion of the simple model of interaction based on the equation array: z q +Ω^2 q -Ω^2<l|_2 u> =w_1, z u +4γ_cδ_{α,x_0}q -Bu +4γ_cδ_{α,x_0}<l|_2 u> =w_2. Besides, less detailed discussion of related models. Central mathematical points: Finite Rank Perturbations Methods, Resolvents formulae, Donoghue-like models, Friedrichs-like models. Central ph…
We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrodinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrodinger group in terms of these resonances.
We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of de Sitter-Reissner-Nordström black hole in terms of resonances. By means of this method we describe the decay of local energy for compactly supported data. The proof uses the cut-off resolvent estimates for the semi-classical Schrödinger operators from Bony and Häfner, 2008. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.
Analysis of leaky modes or wavenumber resonances for the Rayleigh system in a half space
本文受地震学启发，通过引入Jost解、边界矩阵和反射矩阵，全面分析了半空间非均匀板层中Rayleigh算子的波数共振或泄漏模式，刻画了其解析性质与分布，并证明共振是预解式亚纯延拓的极点。
Resonances for Schrödinger operators on infinite cylinders and other products
本文研究 $\mathbb{R}^d\times\mathbb{S}^1$ 上紧支位势 Schrödinger 算子的高能量共振，证明其可由圆平均位势及 $\mathbb{R}^d$ 上的共振刻画，并给出局域化、预解逼近与 $d=1$ 时的渐近和刚性结果。
Continuum limit of resonances for discrete Schrödinger operators
本文证明离散 Schrödinger 算子的复共振在连续极限下收敛到连续算子的共振，并借助推广的范数预解收敛与 Fourier 空间复畸变方法，为奇异势情形提供近似离散算子构造方案。
The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions $d \geq 2$. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto $Λ$, the Riemann surface of the logarithm function. We relate the multiplicities of the poles of the continued scattering matrix to the multiplicities of the poles of the resolvent. Moreover, we show that the poles of the scattering matrix on the $m$th sheet of $Λ$ are related to the zeros of a scalar function defined on the physical sheet. This paper contains a n…
Quantum resonances and scattering poles of classical rank one locally symmetric spaces
本文将负曲率对称空间中散射矩阵极点的已知结论（即交织算子极点或量子共振），利用Bunke和Olbrich的结果推广到具有凸紧基本群的负曲率经典局部对称空间，仅排除对应奇异泊松变换的谱参数。
We reconstruct the rank-one, singular (point-like) perturbations of the $d$-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schrödinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both the possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schrödinger operators formed by a fractional Laplacian and a re…
This paper communicates recent results in theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schrödinger operators. In particular, we propose a formula for computing the norm of a compact complex symmetric operator. This observation is applied to two concrete problems related to quantum mechanical systems. First, we give sharp estimates on the exponential decay of the resolvent and the single-particle density matrix for Schrödinger operators with spectral gaps. Second, we provide new ways of evaluating the resolvent nor…
In resonance to a recent geometric framework proposed by Douglas and Yang, a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space is developed. By taking advantage of the refined existing theory of the principal function of a hyponormal operator we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. A natural field theory interpretation o…
In this paper we provide a detailed description of the eigenvalue $ E_{D}(x_0)\leq 0$ (respectively $ E_{N}(x_0)\leq 0$) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution $-λδ(x-x_0)$ for any fixed value of the magnitude of the coupling constant. We also investigate the $λ$-dependence of both eigenvalues for any fixed value of $x_0$. Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of t…
We show how to find the coefficient by the leading term of the resonance asymptotics using the method of pseudo orbit expansion for quantum graphs which do not obey the Weyl asymptotics. For a non-Weyl graph we develop a method how to reduce the number of edges of a corresponding directed graph. Through this method we prove bounds on the above coefficient depending on the structure of the graph for graphs with the same lengths of internal edges. We explicitly find the positions of the resolvent resonances.
The classical Weyl-von Neumann theorem states that for any self-adjoint operator $A$ in a separable Hilbert space $\mathfrak H$ there exists a (non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed operator $A+C$ has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator $A$ in $\mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $\widetilde A =…
Consider the minimal Sturm-Liouville operator $A = A_{\rm min}$ generated by the differential expression $\mathcal{A} := -\frac{d^2}{dt^2} + T$ in the Hilbert space $L^2(\mathbb{R}_+,\mathcal{H})$ where $T = T^*\ge 0$ in $\mathcal{H}$. We investigate the absolutely continuous parts of different self-adjoint realizations of $\mathcal{A}$. In particular, we show that Dirichlet and Neumann realizations, $A^D$ and $A^N$, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if $\infσ_{ess}(T) = \infσ(T) \ge 0$, then…
Decay estimates for fourth-order Schrödinger operators in dimension two
本文研究了二维有界衰减势下四阶Schrödinger算子的衰减估计，通过预解式在零阈值附近的渐近展开，结合Littlewood-Paley分解与振荡积分理论，建立了$L^1-L^\infty$衰减估计并完成了零共振分类。
Uniform resolvent estimates for magnetic operators
本文研究了磁Schrödinger算子的均匀共振估计，证明了在合适衰减条件下，对于所有复数z，给出了L^p到L^q的估计，并扩展了已知的电磁估计范围。
We give a quantitative version of Vainberg's method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size $τ$ with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate $τ$. Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate $τ$, then there are resonances in logarithmic strips whose width is given by $τ$. As our main application of these results, we give sharp …
This paper is mainly devoted to study time decay estimates of the higher-order Schrödinger type operator $H=(-Δ)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive the asymptotic expansions of resolvent $R_{V}(z)$ near zero threshold with the presence of zero resonance or zero eigenvalue, as well identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen type estimates and local decay estimates for higher order Schrödinger propagator $e^{-itH}$ i…
We study localization effects of disorder on the spectral and dynamical properties of Schroedinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift a…
In this article we prove that for a large class of operators, including Schroedinger operators, with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rate is bounded from below if the classical repeller is sufficiently filamentary. The higher dimensional statement is given in terms of the topological pressure. Under the same assumptions we also prove a resolvent estimate with a logarithmic loss compared to nontrapping estimates.
We discuss the structure of the two- and three-body T-matrices, scattering matrices, and resolvents continued to the unphysical energy sheets. Our conclusions arise due to the representations that have been found for analytically continued momentum-space kernels of the T-operators. These representations are explicitly written only in terms of the physical-sheet kernels of the T-matrix itself. One of advantages of the representations in the three-body case is that they show which portions of the physical-sheet three-body scattering matrix are ``responsible'' for the resonances associated with a…
This paper is devoted to study the time decay estimates for bi-Schrödinger operators $H=Δ^{2}+V(x)$ in dimension one with decaying potentials $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ at zero energy threshold without/with the presence of resonances, and then characterize these resonance spaces corresponding to different types of zero resonance in suitable weighted spaces $L_s^2({\mathbf{R}})$. Next we use them to establish the sharp $L^1-L^\infty$ decay estimates of Schrödinger groups $e^{-itH}$ generated by bi-Schrödinger operators also with zero resonances. As a c…
One-dimensional Schrödinger operators with singular perturbed magnetic and electric potentials are considered. We study the strong resolvent convergence of two families of the operators with potentials shrinking to a point. Localized $δ$-like magnetic fields are combined with $δ\,'$-like perturbations of the electric potentials as well as localized rank-two perturbations. The limit results obtained heavily depend on zero-energy resonances of the electric potentials. In particular, the approximation for a wide class of point interactions in one dimension is obtained.
Recent studies have established that, in addition to the well-known kicked Harper model (KHM), an on-resonance double kicked rotor model (ORDKR) also has Hofstadter's butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter's spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that the quasi-energy spectra of these two dynamical models (i) can exactly overlap with each other if an effective Planck constant takes irrational multiples of 2*pi and (ii) will be different if the same parameter takes rational multiples of 2*pi. This w…
We define a distance between energy forms on a graph-like metric measure space and on a discrete weighted graph using the concept of quasi-unitary equivalence. We apply this result to metric graphs and graph-like manifolds (e.g. a small neighbourhood of an embedded metric graph) as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary) and express the distance of the associated energy forms in terms of geometric quantities. We showed in arXiv:1704.00064 that the ap…
For a class of quantized open chaotic systems satisfying a natural dynamical assumption, we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories.
Two Languages for the Same Resonance: From Nuclear Decay to Black-Hole Ringdown
本文重构量子共振与黑洞QNM等价性的历史谱系，梳理二者从核衰变、黑洞微扰到散射-共振理论的发展脉络，提供对应概念对照表，明确区分数学等价性与历史谱系。
Solvable relaxation in discrete unitary systems: Ruelle-Pollicott resonances and CMV matrices
该研究针对离散幺正系统，以CMV矩阵为对象，通过正交多项式理论识别出两类弛豫相，明确了RP共振的特性及不同方法的一致性，为幺多体系统动力学提供了精确结果。
We consider a Dirac operator with a dislocation potential on the real line. The dislocation potential is a fixed periodic potential on the negative half-line and the same potential but shifted by real parameter $t$ on the positive half-line. Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each non-empty gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that it has only two simple poles on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). Th…
In this work, we are concerned with the mathematical modeling of the electromagnetic (EM) scattering by arbitrarily shaped non-magnetic nanoparticles with high refractive indices. When illuminated by visible light, such particles can exhibit a very strong isotropic magnetic response, resulting from the coupling of the incident wave with the circular displacement currents of the EM fields. The main aim of this work is to mathematically illustrate this phenomenon. We shall first introduce the EM scattering resolvent and the concept of dielectric subwavelength resonances. Then we derive the a pri…
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