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Register a new userDouble operator integral methods applied to continuity of spectral shift functions
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We derive two main results: First, assume that $A$, $B$, $A_n$, $B_n$ are self-adjoint operators in the Hilbert space $mathcal{H}$, and suppose that $A_n$ converges to $A$ and $B_n$ to $B$ in strong resolvent sense as $n to infty$. Fix $m in mathbb{N}$, $m$ odd, $p in [1,infty)$, and assume that $T:= big[( A + iI_{mathcal{H}})^{-m} - ( B + iI_{mathcal{H}})^{-m}big] in mathcal{B}_p(mathcal{H})$, $T_n := big[( A_n + iI_{mathcal{H}})^{-m} - ( B_n + iI_{mathcal{H}})^{-m}big] in mathcal{B}_p(mathcal{H})$, and $lim_{n rightarrow infty} |T_n - T|_{mathcal{B}_p(mathcal{H})} =0$. Then for any function $f$ in the class $mathfrak F_{k}(mathbb{R}) supset C_0^{infty}(mathbb{R})$ (cf. (1.1)), $$ lim_{n rightarrow infty} big| [f(A_n) - f(B_n)] - [f(A)- f(B)]big|_{mathcal{B}_p(mathcal{H})}=0. $$ Our second result concerns the continuity of spectral shift functions $xi(cdot; B,B_0)$ with respect to the operator parameter $B$. For $T$ self-adjoint in $mathcal{H}$ we denote by $Gamma_m(T)$, $m in mathbb{N}$ odd, the set of all self-adjoint operators $S$ in $mathcal{H}$ satisfying $big[(S - z I_{mathcal{H}})^{-m} - (T - z I_{mathcal{H}})^{-m}big] in mathcal{B}_1(mathcal{H})$, $z in mathbb{C}backslash mathbb{R}$. Employing a suitable topology on $Gamma_m(T)$ (cf. (1.9), we prove the following: Suppose that $B_1in Gamma_m(B_0)$ and let ${B_{tau}}_{tauin [0,1]}subset Gamma_m(B_0)$ denote a path from $B_0$ to $B_1$ in $Gamma_m(B_0)$ depending continuously on $tauin [0,1]$ with respect to the topology on $Gamma_m(B_0)$. If $f in L^{infty}(mathbb{R})$, then $$ lim_{tauto 0^+} |xi(, cdot , ; B_{tau}, A_0) f - xi(, cdot , ; B_0, A_0) f|_{L^1(mathbb{R}; (| u|^{m+1} + 1)^{-1}d u)} = 0. $$
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In this paper I prove existence of an irreducible pair of operators $H$ and $H+V,$ where $H$ is a self-adjoint operator and $V$ is a self-adjoint trace-class operator, such that the singular spectral shift function of the pair is non-zero on the absolutely continuous spectrum of the operator $H.$
Given a self-adjoint operator H, a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure is defined such that for all points of this set the wave and the scattering matrices can be defined and constructed unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This new abstract scattering theory allows to prove that for any trace class perturbations of arbitrary self-adjoint operators the singular part of the spectral shift function is an almost everywhere integer-valued function.
We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $lambda^circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be along the eigenfunction $f$, namely $K_0f=0$. The eigenvalue $lambda^circ$ belongs to the spectra of both $H_0$ and $S$. Let $S$ have $sigma$ more eigenvalues below $lambda^circ$ than $H_0$; $sigma$ is known as the spectral shift at $lambda^circ$. We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue $lambda^circ$ in the spectrum of $H(K)=S + K^* K$. We show that the eigenvalue as a function of $K$ has a critical point at $K=K_0$ and the Morse index of this critical point is the spectral shift $sigma$. A version of this theorem also holds for some non-positive perturbations.
This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincare operator. The very notion of spectral analysis has evolved along this path. Indeed, the quest for solving this specific singular integral equation, originally aimed at elucidating classical potential theory problems, has inspired and shaped the development of theoretical spectral analysis of linear transforms in XX-th century. We briefly touch some marking discoveries into the subject, with ample bibliographical references to both old, sometimes forgotten, texts and new contributions. It is remarkable that applications of the spectral analysis of the Neumann-Poincare operator are still uncovered nowadays, with spectacular impacts on applied science. A few modern ramifications along these lines are depicted in our survey.
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.
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