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The Limiting Absorption Principle for Massless Dirac Operators, Properties of Spectral Shift Functions, and an Application to the Witten Index of Non-Fredholm Operators

  • Alan Carey

    Australian National University, Canberra; and University of Wollongong, Australia

  • Fritz Gesztesy

    Baylor University, Waco, USA

  • Galina Levitina

    Australian National University, Canberra, Australia

  • Roger Nichols

    The University of Tennessee at Chattanooga, USA

  • Fedor Sukochev

    University of New South Wales, Sydney, Australia

  • Dmitriy Zanin

    University of New South Wales, Sydney, Australia
The Limiting Absorption Principle for Massless Dirac Operators, Properties of Spectral Shift Functions, and an Application to the Witten Index of Non-Fredholm Operators cover
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Applying the theory of strongly smooth operators, we derive a limiting absorption principle (LAP) on any compact interval in R\{0}\mathbb{R} \backslash \{0\} for the free massless Dirac operator,

H0=α⋅(−i∇)H_0 = \alpha \cdot (-i \nabla)

in [L2(Rn)]N[L^2(\mathbb{R}^n)]^N, n∈Nn \in \mathbb{N}, n≥2n \geq 2, N=2⌊(n+1)/2⌋N=2^{\lfloor(n+1)/2\rfloor}. We then use this to demonstrate the absence of singular continuous spectrum of interacting massless Dirac operators H=H0+VH = H_0 +V, where the entries of the (essentially bounded) matrix-valued potential VV decay like O(∣x∣−1−ε){O}\big(|x|^{-1 - \varepsilon}\big) as ∣x∣→∞|x| \to \infty for some ε>0\varepsilon > 0. This includes the special case of electromagnetic potentials decaying at the same rate. In addition, we derive a one-to-one correspondence between embedded eigenvalues of HH in R\{0}\mathbb{R} \backslash \{0\} and the eigenvalue −1- 1 of the (normal boundary values of the) Birman–Schwinger-type operator

V2(H0−(λ0±i0)I[L2(Rn)]N)−1V1∗‾.\overline{V_2 \big(H_0 - (\lambda_0 \pm i 0)I_{[L^2(\mathbb{R}^n)]^N}\big)^{-1} V_1^*}.

Upon expressing ξ(⋅ ;H,H0)\xi(\cdot\,; H,H_0) as normal boundary values of regularized Fredholm determinants to the real axis, we then prove that in the concrete case (H,H0)(H,H_0), under appropriate hypotheses on VV (implying the decay of VV like O(∣x∣−n−1−ε){O}\big(|x|^{-n -1 - \varepsilon}\big) as ∣x∣→∞|x| \to \infty), the associated spectral shift function satisfies ξ(⋅ ;H,H0)∈C((−∞,0)∪(0,∞))\xi(\cdot\,;H,H_0) \in C((-\infty,0) \cup (0,\infty)), and that the left and right limits at zero, ξ(0±;H,H0)=lim⁡ε↓0ξ(±ε;H,H0)\xi(0_{\pm}; H,H_0) = \lim_{\varepsilon \downarrow 0} \xi(\pm \varepsilon; H,H_0), exist.

This fact is then used to express the resolvent regularized Witten index of the non-Fredholm operator DA\boldsymbol{D}_{\boldsymbol{A}}^{} in L2(R;[L2(Rn)]N)L^2\big(\mathbb{R};[L^2(\mathbb{R}^n)]^N\big) given by

DA=ddt+A,dom⁡(DA)=W1,2(R;[L2(Rn)]N)∩dom⁡(A−),{\boldsymbol{D}}_{\boldsymbol{A}}^{} = \frac{d}{dt} + \boldsymbol{A}, \quad \operatorname{dom}(\boldsymbol{D}_{\boldsymbol{A}}^{})= W^{1,2}\big(\mathbb{R}; [L^2(\mathbb{R}^n)]^N\big) \cap \operatorname{dom}(\boldsymbol{A}_-),

where

A=A−+B,dom⁡(A)=dom⁡(A−),\boldsymbol{A} = \boldsymbol{A}_- + \boldsymbol{B}, \quad \operatorname{dom}(\boldsymbol{A}) = \operatorname{dom}(\boldsymbol{A}_-),

in terms of ξ(⋅ ;H,H0)\xi(\cdot\,; H,H_0). Here A\boldsymbol{A}, A−\boldsymbol{A}_-, A+\boldsymbol{A}_+, B\boldsymbol{B}, and B+\boldsymbol{B}_+ in L2(R;[L2(Rn)]N)L^2\big(\mathbb{R};[L^2(\mathbb{R}^n)]^N\big) are generated with the help of the Dirac-type operators H,H0H, H_0 and potential matrices VV, via

A(t)=A−+B(t),  t∈R,A−=H0,A+=A−+B+=H,B(t)=b(t)B+,  t∈R,B+=V,\begin{align*} & A(t) = A_- + B(t), \; t \in \mathbb{R}, \quad A_- = H_0, \quad A_+ = A_- + B_+ = H, \\ & B(t)=b(t) B_+, \; t \in \mathbb{R}, \quad B_+ = V, \end{align*}

in [L2(Rn)]N[L^2(\mathbb{R}^n)]^N, assuming

b(k)∈C∞(R)∩L∞(R;dt),  k∈N0,b′∈L1(R;dt),lim⁡t→∞b(t)=1,lim⁡t→−∞b(t)=0.\begin{align*} & b^{(k)} \in C^{\infty}(\mathbb{R}) \cap L^{\infty}(\mathbb{R}; dt), \; k \in \mathbb{N}_0, \quad b' \in L^1(\mathbb{R}; dt), \\ & \lim_{t \to \infty} b(t) = 1, \quad \lim_{t \to - \infty} b(t) = 0. \end{align*}

In particular, A±A_{\pm} are the asymptotes of the family A(t)A(t), t∈Rt \in \mathbb{R}, as t→±∞t \to \pm \infty in the norm resolvent sense. (Here L2(R;H)=∫R⊕dt HL^2(\mathbb{R};\mathcal{H}) = \int_{\mathbb{R}}^{\oplus} dt \, \mathcal{H} and T=∫R⊕dt T(t)\boldsymbol{T} = \int_{\mathbb{R}}^{\oplus} dt \, T(t) represent direct integrals of Hilbert spaces and operators.)

Introducing the nonnegative, self-adjoint operators

H1=DA∗DA,H2=DADA∗\boldsymbol{H}_1 = \boldsymbol{D}_{\boldsymbol{A}}^{*} \boldsymbol{D}_{\boldsymbol{A}}^{}, \quad \boldsymbol{H}_2 = \boldsymbol{D}_{\boldsymbol{A}}^{} \boldsymbol{D}_{\boldsymbol{A}}^{*}

in L2(R;[L2(Rn)]N)L^2\big(\mathbb{R};[L^2(\mathbb{R}^n)]^N\big), one of the principal results proved in this manuscript expresses the resolvent regularized Witten index Wk,r(DA)W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}}^{}) of DA\boldsymbol{D}_{\boldsymbol{A}}^{} in terms of spectral shift functions via

Wk,r(DA)=ξL(0+;H2,H1)=[ξ(0+;H,H0)+ξ(0−;H,H0)]/2,k∈N,  k≥⌈n/2⌉.W_{k,r}(\boldsymbol{D}_{\boldsymbol{A}}^{}) = \xi_L(0_+; \boldsymbol{H}_2, \boldsymbol{H}_1) = \big[\xi(0_+;H,H_0) + \xi(0_-;H,H_0)\big]\big/2, \quad k \in \mathbb{N}, \; k \geq \lceil n\big/2 \rceil.

Here the notation ξL(0+;H2,H1)\xi_L(0_+; \boldsymbol{H}_2, \boldsymbol{H}_1) indicates that 00 is a right Lebesgue point for ξ(⋅ ;H2,H1)\xi(\cdot\,; \boldsymbol{H}_2, \boldsymbol{H}_1), and Wk,r(T)W_{k,r}(T) represents the kkth resolvent regularized Witten index of the densely defined, closed operator TT in the complex, separable Hilbert space K\mathcal{K}, defined by

Wk,r(T)=lim⁡λ↑0(−λ)ktr⁡K((T∗T−λIK)−k−(T T∗−λIK)−k),W_{k,r}(T) = \lim_{\lambda \uparrow 0} (- \lambda)^k \operatorname{tr}_{\mathcal{K}} \big((T^* T - \lambda I_{\mathcal{K}})^{-k} - (T\,T^* - \lambda I_{\mathcal{K}})^{-k}\big),

whenever the limit exists for some k∈Nk \in \mathbb{N}.