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Register a new userOn the relationship of spectral flow to the Fredholm index and its extension to non-Fredholm operators
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In cite{APSIII} Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin-Salamon cite{RS95}. In cite{GLMST}, a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper cite{Pu08}. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and {its perturbation by a relatively trace-class operator}. In this paper we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher $p^{th}$ Schatten class condition for $0leq p<infty$, thus allowing differential operators on manifolds of any dimension $d<p+1$. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by cite{BCPRSW, CGK16}. This leads us to introduce a notion of `generalised spectral flow for such paths and to investigate its properties.
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An analytic definition of a $mathbb{Z}_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings through $0$ along the path. The $mathbb{Z}_2$-valued spectral flow is shown to satisfy a concatenation property and homotopy invariance, and it provides an isomorphism on the fundamental group of the real skew-adjoint Fredholm operators. Moreover, it is connected to a $mathbb{Z}_2$-index pairing for suitable paths. Applications concern the zero energy bound states at defects in a Majorana chain and a spectral flow interpretation for the $mathbb{Z}_2$-polarization in these models.
In this article we give a comprehensive treatment of a `Clifford module flow along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in KO${}_{*}(mathbb{R})$ via the Clifford index of Atiyah-Bott-Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that [ text{spectral flow} = text{Fredholm index}. ] That is, we show how the KO--valued spectral flow relates to a KO-valued index by proving a Robbin-Salamon type result. The Kasparov product is also used to establish a spectral flow $=$ Fredholm index result at the level of bivariant K-theory. We explain how our results incorporate previous applications of $mathbb{Z}/ 2mathbb{Z}$-valued spectral flow in the study of topological phases of matter.
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.
Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P in psi^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i to M$, $i = 0,1$, and let $alpha$ be an irreducible representation of the group $G$. Then $P$ induces a map $pi_alpha(P) : H^s(M; E_0)_alpha to H^{s-m}(M; E_1)_alpha$ between the $alpha$-isotypical components. We prove that the map $pi_alpha(P)$ is Fredholm if, and only if, $P$ is {em transversally $alpha$-elliptic}, a condition defined in terms of the principal symbol of $P$ and the action of $G$ on the vector bundles $E_i$.
Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from $(1+1)$-dimensional differential operators using the model operator $D_A$ in $L^2(mathbb{R}^2; dt dx)$ of the type $D_A = (d/dt) + A$, where $A = int^{oplus}_{mathbb{R}} dt , A(t)$, and the family of self-adjoint operators $A(t)$ in $L^2(mathbb{R}; dx)$ is explicitly given by $A(t) = - i (d/dx) + theta(t) phi(cdot)$, $t in mathbb{R}$. Here $phi: mathbb{R} to mathbb{R}$ has to be integrable on $mathbb{R}$ and $theta: mathbb{R} to mathbb{R}$ tends to zero as $t to - infty$ and to $1$ as $t to + infty$. In particular, $A(t)$ has asymptotes in the norm resolvent sense $A_- = - i (d/dx)$, $A_+ = - i (d/dx) + phi(cdot)$ as $t to mp infty$. Since $D_A$ violates the relative trace class condition introduced in [9], we now employ a new approach based on an approximation technique. The approximants do fit the framework of [9] and lead to the following results: Introducing $H_1 = {D_A}^* D_A$, $H_2 = D_A {D_A}^*$, we recall that the resolvent regularized Witten index of $D_A$, denoted by $W_r(D_A)$, is defined by $$ W_r(D_A) = lim_{lambda to 0} (- lambda) {rm tr}_{L^2(mathbb{R}^2; dtdx)}((H_1 - lambda I)^{-1} - (H_2 - lambda I)^{-1}). $$ In the concrete example at hand, we prove $$ W_r(D_A) = xi(0_+; H_2, H_1) = xi(0; A_+, A_-) = 1/(2 pi) int_{mathbb{R}} dx , phi(x). $$ Here $xi(, cdot , ; S_2, S_1)$, denotes the spectral shift operator for the pair $(S_2,S_1)$, and we employ the normalization, $xi(lambda; H_2, H_1) = 0$, $lambda < 0$.
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