2021/05/07 by Carey, Alan, Gesztesy, Fritz, Levitina, Galina +3 · 3 citations
#35Q40 #47A40 #81Q10 #FOS: Mathematics #Primary: 35P25 #Secondary: 47A10 #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2105.03024
We derive a limiting absorption principle on any compact interval in ℝ \backslash \0\ for the free massless Dirac operator, H0 = α⋅ (-i ∇) in [L2(ℝn)]N, n ≥ 2, N=2\lfloor(n+1)/2\rfloor, and then prove the absence of singular continuous spectrum of interacting massless Dirac operators H = H0 +V, where V decays like O(|x|-1 - ε). Expressing the spectral shift function ξ( ⋅ ; H,H0) as normal boundary values of regularized Fredholm determinants, we prove that for sufficiently decaying V, ξ( ⋅ ;H,H0) ∈ C((-∞,0) ∪ (0,∞)), and that the left and right limits at zero, ξ(0±; H,H0), exist. Introducing the non-Fredholm operator \boldsymbolD_\boldsymbolA = (d)/(dt) + \boldsymbolA in L2(ℝ;[L2(ℝn)]N), where \boldsymbolA = \boldsymbolA- + \boldsymbolB, \boldsymbolA-, and \boldsymbolB are generated in terms of H, H0 and V, via A(t) = A- + B(t), A- = H0, B(t)=b(t) V, t ∈ ℝ, assuming b is smooth, b(-∞) = 0, b(+∞) = 1, and introducing \boldsymbolH1 = \boldsymbolD_\boldsymbolA* \boldsymbolD_\boldsymbolA, \boldsymbolH2 = \boldsymbolD_\boldsymbolA \boldsymbolD_\boldsymbolA*, one of the principal results in this manuscript expresses the kth resolvent regularized Witten index Wk,r(\boldsymbolD_\boldsymbolA) (k ∈ ℕ, k ≥ \lceil n/2 \rceil) in terms of spectral shift functions as Wk,r(\boldsymbolD_\boldsymbolA) = ξ(0+; \boldsymbolH2, \boldsymbolH1) = [ξ(0+;H,H0) + ξ(0-;H,H0)]/2. Here L2(ℝ;H) = ∫ℝ⊕ dt H and \boldsymbolT = ∫ℝ⊕ dt T(t) abbreviate direct integrals.