Spectral non-self-adjoint analysis of complex Dirac, Pauli and Schr\"odinger operators of full rank with constant magnetic fields

Abstract

We consider Dirac, Pauli and Schr\"odinger quantum magnetic Hamiltonians of full rank in ${\rm L}^2 \big(\mathbb{R}^{2d} \big)$, $d \ge 1$, perturbed by non-self-adjoint (matrix-valued) potentials. On the one hand, we show the existence of non-self-adjoint perturbations, generating near each point of the essential spectrum of the operators, infinitely many (complex) eigenvalues. In particular, we establish point spectrum analogous of B\"ogli results [B\"og17] obtained for non-magnetic Laplacians, and hence showing that classical Lieb-Thirring inequalities cannot hold for our magnetic models. On the other hand, we give asymptotic behaviours of the number of the (complex) eigenvalues. In particular, for compactly supported potentials, our results establish non-self-adjoint extensions of Raikov-Warzel [RW02] and Melgaard-Rozenblum [MR03] results. So, we show how the (complex) eigenvalues converge to the points of the essential spectrum asymptotically, i.e., up to a multiplicative explicit constant, as $$ \frac{1}{d!} \Bigg(\frac{\vert \ln r \vert}{\ln \vert \ln r \vert} \Bigg)^d, \quad r \searrow 0, $$ in small annulus of radius $r > 0$ around the points of the essential spectrum.