Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPlasmonic eigenvalue problem for corners: limiting absorption principle and absolute continuity in the essential spectrum
69
0
0.0
(
0
)
Authors
Karl-Mikael Perfekt
Ask ChatGPT about the research
No Arabic abstract
We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann--Poincare operator of the boundary. A limiting absorption principle is proved, valid when the spectral parameter approaches the essential spectrum. Putting the principle into use, it is proved that the corner produces absolutely continuous spectrum of multiplicity 1. The embedded eigenvalues are discrete. In particular, there is no singular continuous spectrum.
rate research
Read More
Two types of eigenvalue continuity are commonly used in the literature. However, their meanings and the conditions under which continuities are used are not always stated clearly. This can lead to some confusion and needs to be addressed. In this note, we revisit the Gerv{s}gorin disk theorem and clarify the issue concerning the proofs of the theorem by continuity.
Given a self-adjoint operator $H_0$ and a relatively $H_0$-compact self-adjoint operator $V,$ the functions $r_j(z) = - sigma_j^{-1}(z),$ where $sigma_j(z)$ are eigenvalues of the compact operator $(H_0-z)^{-1}V,$ bear a lot of important information about the pair $H_0$ and $V.$ We call them coupling resonances. In case of rank one (and positive) perturbation $V,$ there is only one coupling resonance function, which is a Herglotz function. This case has been studied in depth in the literature, and appears in different situations, such as Sturm-Liouville theory, random Schrodinger operators, harnomic and spectral analyses, etc. The general case is complicated by the fact that the resonance functions are no longer single valued holomorphic functions, and potentially can have quite an erratic behaviour, typical for infinitely-valued holomorphic functions. Of special interest are those coupling resonance functions $r_z$ which approach a real number $r_{lambda+i0}$ from the interval $[0,1]$ as the spectral parameter $z=lambda+iy$ approaches a point $lambda$ of the essential spectrum, since they are responsible for spectral flow through $lambda$ inside essential spectrum when $H_0$ gets deformed to $H_1 = H_0+V$ via the path $H_0 + rV, r in [0,1].$ In this paper it is shown that if the pair $H_0,$ $V$ satisfies the limiting absorption principle, then the coupling resonance functions are well-behaved near the essential spectrum in the following sense. Let $I$ be an open interval inside the essential spectrum of $H_0$ and $epsilon>0.$ Then there exists a compact subset~$K$ of~$I$ such that $| I setminus K | < epsilon,$ and $K$ has a non-tangential neighbourhood in the upper complex half-plane, such that any coupling resonance function is either single-valued in the neighbourhood, or does not take a real value in the interval $[0,1].$
The spectral flow is a classical notion of functional analysis and differential geometry which was given different interpretations as Fredholm index, Witten index, and Maslov index. The classical theory treats spectral flow outside the essential spectrum. Inside essential spectrum, the spectral shift function could be considered as a proper analogue of spectral flow, but unlike the spectral flow, the spectral shift function is not an integer-valued function. In this paper it is shown that the notion of spectral flow admits a natural integer-valued extension for a.e. value of the spectral parameter inside essential spectrum too and appropriate theory is developed. The definition of spectral flow inside essential spectrum given in this paper applies to the classical spectral flow and thus gives one more new alternative definition of it.
We derive a limiting absorption principle on any compact interval in $mathbb{R} backslash {0}$ for the free massless Dirac operator, $H_0 = alpha cdot (-i abla)$ in $[L^2(mathbb{R}^n)]^N$, $n geq 2$, $N=2^{lfloor(n+1)/2rfloor}$, and then prove the absence of singular continuous spectrum of interacting massless Dirac operators $H = H_0 +V$, where $V$ decays like $O(|x|^{-1 - varepsilon})$. Expressing the spectral shift function $xi(,cdot,; H,H_0)$ as normal boundary values of regularized Fredholm determinants, we prove that for sufficiently decaying $V$, $xi(,cdot,;H,H_0) in C((-infty,0) cup (0,infty))$, and that the left and right limits at zero, $xi(0_{pm}; H,H_0)$, exist. Introducing the non-Fredholm operator $boldsymbol{D}_{boldsymbol{A}} = frac{d}{dt} + boldsymbol{A}$ in $L^2big(mathbb{R};[L^2(mathbb{R}^n)]^Nbig)$, where $boldsymbol{A} = boldsymbol{A_-} + boldsymbol{B}$, $boldsymbol{A_-}$, and $boldsymbol{B}$ are generated in terms of $H, H_0$ and $V$, via $A(t) = A_- + B(t)$, $A_- = H_0$, $B(t)=b(t) V$, $t in mathbb{R}$, assuming $b$ is smooth, $b(-infty) = 0$, $b(+infty) = 1$, and introducing $boldsymbol{H_1} = boldsymbol{D}_{boldsymbol{A}}^{*} boldsymbol{D}_{boldsymbol{A}}$, $boldsymbol{H_2} = boldsymbol{D}_{boldsymbol{A}} boldsymbol{D}_{boldsymbol{A}}^{*}$, one of the principal results in this manuscript expresses the $k$th resolvent regularized Witten index $W_{k,r}(boldsymbol{D}_{boldsymbol{A}})$ ($k in mathbb{N}$, $k geq lceil n/2 rceil$) in terms of spectral shift functions as [ W_{k,r}(boldsymbol{D}_{boldsymbol{A}}) = xi(0_+; boldsymbol{H_2}, boldsymbol{H_1}) = [xi(0_+;H,H_0) + xi(0_-;H,H_0)]/2. ] Here $L^2(mathbb{R};mathcal{H}) = int_{mathbb{R}}^{oplus} dt , mathcal{H}$ and $boldsymbol{T} = int_{mathbb{R}}^{oplus} dt , T(t)$ abbreviate direct integrals.
This paper studies the size of the minimal gap between any two consecutive eigenvalues in the Dirichlet and in the Neumann spectrum of the standard Laplace operator on the Sierpinski gasket. The main result shows the remarkable fact that this minimal gap is achieved and coincides with the spectral gap. The Dirichlet case is more challenging and requires some key observations in the behavior of the dynamical system that describes the spectrum.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
