Applying the theory of strongly smooth operators, we derive a limiting absorption principle (LAP) on any compact interval in R\{0} for the free massless Dirac operator,
H0=α⋅(−i∇)
in [L2(Rn)]N, n∈N, n≥2, N=2⌊(n+1)/2⌋. We then use this to demonstrate the absence of singular continuous spectrum of interacting massless Dirac operators H=H0+V, where the entries of the (essentially bounded) matrix-valued potential V decay like O(∣x∣−1−ε) as ∣x∣→∞ for some ε>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 H in R\{0} and the eigenvalue −1 of the (normal boundary values of the) Birman–Schwinger-type operator
V2(H0−(λ0±i0)I[L2(Rn)]N)−1V1∗.
Upon expressing ξ(⋅;H,H0) as normal boundary values of regularized Fredholm determinants to the real axis, we then prove that in the concrete case (H,H0), under appropriate hypotheses on V (implying the decay of V like O(∣x∣−n−1−ε) as ∣x∣→∞), the associated spectral shift function satisfies ξ(⋅;H,H0)∈C((−∞,0)∪(0,∞)),
and that the left and right limits at zero, ξ(0±;H,H0)=limε↓0ξ(±ε;H,H0), exist.
This fact is then used to express the resolvent regularized Witten index of the non-Fredholm operator
DA in L2(R;[L2(Rn)]N) given by
DA=dtd+A,dom(DA)=W1,2(R;[L2(Rn)]N)∩dom(A−),
where
A=A−+B,dom(A)=dom(A−),
in terms of ξ(⋅;H,H0). Here A, A−, A+, B, and B+ in L2(R;[L2(Rn)]N) are generated with the help of the Dirac-type operators H,H0 and potential matrices V, via
A(t)=A−+B(t),t∈R,A−=H0,A+=A−+B+=H,B(t)=b(t)B+,t∈R,B+=V,
in [L2(Rn)]N, assuming
b(k)∈C∞(R)∩L∞(R;dt),k∈N0,b′∈L1(R;dt),t→∞limb(t)=1,t→−∞limb(t)=0.
In particular, A± are the asymptotes of the family A(t), t∈R, as t→±∞ in the norm resolvent sense. (Here L2(R;H)=∫R⊕dtH and
T=∫R⊕dtT(t) represent direct integrals of Hilbert spaces and operators.)
Introducing the nonnegative, self-adjoint operators
H1=DA∗DA,H2=DADA∗
in L2(R;[L2(Rn)]N), one of the principal results proved in this manuscript expresses the resolvent regularized Witten index
Wk,r(DA) of DA 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⌉.
Here the notation ξL(0+;H2,H1) indicates that 0 is a right Lebesgue point for ξ(⋅;H2,H1), and Wk,r(T) represents the kth resolvent regularized Witten index of the densely defined, closed operator T in the complex, separable Hilbert space K, defined by
Wk,r(T)=λ↑0lim(−λ)ktrK((T∗T−λIK)−k−(TT∗−λIK)−k),
whenever the limit exists for some k∈N.