Resonances near thresholds of magnetic Pauli and Dirac operators (Q2871243)

The author studies the number of resonances of some perturbations of the free Pauli operator \(H_0\) and the free Dirac operator \(D_0\) on \(C^\infty_0(\mathbb R^3,\mathbb C^2)\). More precisely, consider the operators NEWLINE\[NEWLINE H:=H_0+V \qquad D:=D_0+V,NEWLINE\]NEWLINE where the electric potential \(V\) is assumed to have super-exponential decay in the direction of the magnetic field. The resolvents of \(H\) and \(D\) have meromorphic extensions to the the upper half-plane; the resonances of \(H\) and \(D\) are defined as the poles of these meromorphic extensions.NEWLINENEWLINE\smallskipNEWLINENEWLINEThe first two main results of the paper are concerned with the number of resonances of \(H\) near the origin. If \(\text{Res}(H)\) denotes the set of resonances of \(H\) near \(0\), then the author shows there exists \(r_0>0\) such that NEWLINE\[NEWLINE \#\{ k^2 \in \text{Res}(H): r<|k|<2r\} = \mathcal O( \varphi(r) ) +\mathcal O(1) \qquad (0<r<r_0),NEWLINE\]NEWLINE where \(\varphi(r)\) is some function of \(r\). The author provides explicit expressions for \(\varphi(r)\) under various conditions. The author then refines the above result for the perturbed operators \(H_e:=H_0+eV\), \(V>0\), where \(e\) is a non-zero real number outside some discrete set. In particular, he shows that NEWLINE\[NEWLINE \#\{ k^2 \in \text{Res}(H_e): r<|k|<2r\} = \psi(r) (1+o(1)) \qquad (r \to 0^+),NEWLINE\]NEWLINE where \(\psi(r)\) can be easily expressed in terms of the function \(\varphi(r)\) above. Similar results for the Dirac operators \(D\) and \(D_e\) are also obtained in a neighborhood of \(\pm m\), where \(m\) is the mass of some particle.