Some new results about the massless Dirac operator

Abstract

In this paper, we give a method to construct the zero mode and zero resonance of the massless Dirac operator H = α · D + Q(x) and obtain some special examples to show the sharpness of some existing results. On the other hand, we extend the previous results of H about the vanishing properties of the zero mode and the existence of the zero resonance. In Sec. IV, we consider the local version and get some improvements and extensions. C 2013 American Institute of Physics.

One-dimensional Coulomb problem in Dirac materials

“…Furthermore, it should be mentioned that the confinement of Dirac-like particles in 1D potentials is the subject of considerable recent attention from the applied mathematics community [41][42][43][44]. Our results here, using two explicit toy models of nonintegrable potentials, provide a complementary approach both more accessible to physicists and closer to experimental reality.…”

Absence of zero resonances of massless Dirac operators

“…We remark that Saitō -Umeda [6] and Zhong -Gao [7] have proven the following result under the same assumption |Q(x)| ≤ C x −ρ , ρ > 1 (In [6], it is assumed ρ > 3/2, however, arguments of [6] go through under the assumption ρ > 1 as was made explicit in [7]): If f satisfies f ∈ L 2,−s (R 3 ) for some 0 < s ≤ min{3/2, ρ − 1} and Hf = 0 in the sense of distributions, then f ∈ H 1 (R 3 ). Our theorem improves over the results of [6] and [7] by weakening the assumption f ∈ L 2,−s (R 3 ) to L 2,−3/2 (R 3 ), which is ρ > 1 independent, and by strengthening the result f ∈ H 1 (R 3 ) to a sharp decay estimate x µ f ∈ H 1 (R 3 ), µ < 1/2. We briefly explain the significance of the theorem.…”

On absence of threshold resonances for Schrödinger and Dirac operators

“…(iii) As mentioned in Remark 2.3, the absence of zero-energy resonances is wellknown in the three-dimensional case n = 3, see [7], [12,Sect. 4.4], [13], [16], [67], [68], [78]. In fact, for n = 3 the absence of zero-energy resonances has been shown under the weaker decay [7].…”

“…This is consistent with observations in [7], [12,Sect. 4.4], [13], [16], [67], [68], [78] for n = 3 (see also Remark 3.9 (iii)). This should be contrasted with the behavior of Schrödinger operators where lim z→0 z∈C\{0} 38) "implies" the absence of zero-energy resonances of h = h 0 + v for n ∈ N, n 5, again for sufficiently fast decaying short-range potentials v at infinity, as | • | 2−n lies in L 2 (R n ; d n x) near infinity if and only if n 5, as observed in [50].…”

“…While a complete bibliography in this connection is clearly beyond the scope of our paper, we refer to [2]- [11], [12,Ch. 4], [13], [15]- [21], [28]- [43], [45], [46], [50]- [54], [58]- [62], [65]- [73], [76]- [78], and the literature therein to demonstrate some of the interest generated by this circle of ideas. We also mention the relevance of threshold states in connection with the Witten index for non-Fredholm operators, see, for instance, [22]- [26].…”