Eigenvalue bounds for non-selfadjoint Dirac operators (Q2163415)

This paper is devoted to the proof of eigenvalue bounds for Dirac operators coupled with complex valued potentials that are small in a suitable sense. To describe the main results, let \(n \geq 2\), let \(N = 2^{[n/2]}\), and let \[ \mathscr{D}_0 = -i \sum_{k=1}^n \alpha_k \frac{\partial}{\partial x_k} + m \alpha_0 \] be the free Dirac operator in \(\mathbb{R}^n\), where \(m \geq 0\) and \(\alpha_0, \dots, \alpha_n \in \mathbb{C}^{N \times N}\) are a family of anti-commuting, unitary, and self-adjoint matrices. Moreover, assume that \(V: \mathbb{R}^n \rightarrow \mathbb{C}^{N \times N}\) is measurable such that \[ \| V \|_Y := \max_{j = 1, \dots, n} \left(\int_{\mathbb{R}} \| V \|_{L^\infty(\mathbb{R}^{n-1}, dx_1 \dots d x_{j-1} d x_{j+1} \dots d x_n)} d x_j \right) < C_0 \] for a sufficiently small constant \(C_0\). Then, the authors prove that in the massive case (\(m>0\)) all eigenvalues of \(\mathscr{D}_0 + V\) are contained in the two closed disks \(\overline{B}_{R_0}(x_0^\pm)\), where \[ x_0^\pm = \pm m \frac{\nu^2+1}{\nu^2-1}, \quad R_0 = m \frac{2 \nu}{\nu^2-1}, \quad \text{and} \quad \nu = \left[ \frac{(n+1) C_0}{\|V\|_Y}-n \right]^2>1. \] Moreover, in the massless case (\(m=0\)) it is shown that \(\mathscr{D}_0 + V\) has no eigenvalues and \(\sigma(\mathscr{D}_0+V)=\mathbb{R}\), provided that \(\| V \|_Y\) is sufficiently small. In order to prove these results, the authors show uniform bounds for the resolvent of \(\mathscr{D}_0\) in a suitable norm and an abstract variant of the Birman-Schwinger principle, that is, in particular, useful to exclude embedded eigenvalues.