Spectral non-self-adjoint analysis of complex Dirac,Pauli and Schrödinger operators with constant magnetic fields of full rank

Abstract

We consider Dirac, Pauli and Schrödinger quantum Hamiltonians with constant magnetic fields of full rank in ${\mathrm{L}}^2\mathrm{(}{\mathbb{R}}^{2\mathit{d}}\mathrm{)}$ , $\mathit{d}⩾1$ , 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. 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 [Rev. in Math. Physics 14 (2002), 1051–1072] and Melgaard–Rozenblum [Commun. PDE. 28 (2003), 697–736] 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 $$\begin{array}{c}{\displaystyle \frac{1}{\mathit{d}!}}{\mathrm{\left(}{\displaystyle \frac{|ln\mathit{r}|}{ln|ln\mathit{r}|}}\mathrm{\right)}}^{\mathit{d}}\mathrm{,}\mathit{r}\searrow 0\mathrm{,}\end{array}$$ in small annulus of radius $\mathit{r}\mathrm{>}0$ around the points of the essential spectrum.