An enhanced term in the Szegő-type asymptotics for the free massless Dirac operator

We consider a regularised Fermi projection of the Hamiltonian of the massless Dirac equation at Fermi energy zero. The matrix-valued symbol of the resulting operator is discontinuous in the origin. For this operator, we prove Szegő-type asymptotics with the spatial cut-off domains given by $d$-dimensional cubes. For analytic test functions, we obtain a $d$-term asymptotic expansion and provide an upper bound of logarithmic order for the remaining terms. This bound does not depend on the regularisation. In the special case that the test function is given by a polynomial of degree less or equal than three, we prove a $(d+1)$-term asymptotic expansion with an error term of constant order. The additional term is of logarithmic order and its coefficient is independent of the regularisation.

💡 Research Summary

The paper investigates Szegő‑type asymptotics for a regularised Fermi projection associated with the free massless Dirac operator in spatial dimensions d ≥ 2. The Dirac Hamiltonian 𝔇 = −i∑_{k=1}^{d}α_k∂k acts on L²(ℝ^d)⊗ℂⁿ, where the α‑matrices satisfy the usual Clifford algebra relations and n = 2⌊(d+1)/2⌋. After Fourier transformation the operator is multiplication by the matrix‑valued symbol D(ξ)=∑{k=1}^{d}α_k ξ_k, which is smooth everywhere except at the origin where the energy |ξ| vanishes.

To obtain a bounded operator one introduces an ultraviolet cut‑off parameter b ≥ 0 and a smooth step function χ⁰_b(x)=1_{x<0} · φ(x+b) with φ a smooth monotone cutoff equal to 0 for x ≤ −1 and 1 for x ≥ 0. The regularised Fermi projection is defined as Op