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Register a new userThe limiting absorption principle for massless Dirac operators, properties of spectral shift functions, and an application to the Witten index of non-Fredholm operators
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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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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.
Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral properties of the perturbed operator $H_0+V$. The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.
Let $H_0 = -Delta + V_0(x)$ be a Schroedinger operator on $L_2(mathbb{R}^ u),$ $ u=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $mathbb{R}^ u.$ Let $V$ be an operator of multiplication by a bounded integrable real-valued function $V(x)$ and put $H_r = H_0+rV$ for real $r.$ We show that the associated spectral shift function (SSF) $xi$ admits a natural decomposition into the sum of absolutely continuous $xi^{(a)}$ and singular $xi^{(s)}$ SSFs. This is a special case of an analogous result for resolvent comparable pairs of self-adjoint operators, which generalises the known case of a trace class perturbation while also simplifying its proof. We present two proofs -- one short and one long -- which we consider to have value of their own. The long proof along the way reframes some classical results from the perturbation theory of self-adjoint operators, including the existence and completeness of the wave operators and the Birman-Krein formula relating the scattering matrix and the SSF. The two proofs demonstrate the equality of the singular SSF with two a priori different but intrinsically integer-valued functions: the total resonance index and the singular $mu$-invariant.
We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described in a previor result of the last two authors with Berkolaiko, where it was used to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the eigenfunctions of these operators. In particular, we give explicit formulas for the limiting eigenfunctions, and also characterize the eigenfunctions and eigenvalues for all values for the spectral flow parameter (not just in the limit). We also develop spectrally accurate numerical tools for comparison and visualization.
We consider harmonic Toeplitz operators $T_V = PV:{mathcal H}(Omega) to {mathcal H}(Omega)$ where $P: L^2(Omega) to {mathcal H}(Omega)$ is the orthogonal projection onto ${mathcal H}(Omega) = left{u in L^2(Omega),|,Delta u = 0 ; mbox{in};Omegaright}$, $Omega subset {mathbb R}^d$, $d geq 2$, is a bounded domain with $partial Omega in C^infty$, and $V: Omega to {mathbb C}$ is a suitable multiplier. First, we complement the known criteria which guarantee that $T_V$ is in the $p$th Schatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V in S_{p, {rm w}}$, the weak counterpart of $S_p$. Next, we assume that $Omega$ is the unit ball in ${mathbb R}^d$, and $V = overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $partial Omega$ or $V$ is compactly supported in $Omega$. Further, we consider general $Omega$ and $V geq 0$ which is regular in $Omega$, and admits a power-like decay of rate $gamma > 0$ at $partial Omega$, and we show that in this case $T_V$ is unitarily equivalent to a pseudo-differential operator of order $-gamma$, self-adjoint in $L^2(partial Omega)$. Using this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for the operator $T_V$. Finally, we introduce the Krein Laplacian $K geq 0$, self-adjoint in $L^2(Omega)$; it is known that ${rm Ker},K = {mathcal H}(Omega)$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V in C(overline{Omega};{mathbb R})$, and show that $sigma_{rm ess}(K+V) = V(partial Omega)$. Assuming that $V geq 0$ and $V{|partial Omega} = 0$, we study the asymptotic distribution of the eigenvalues of $K pm V$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator $T_V$.
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