Exponential decay of eigenfunctions of first order systems

Papers from the International Conference on Transport and Spectral Problems in Quantum Mechanics held in honor of Jean-Michel Combes at the Université de Cergy-Pontoise, Cergy-Pontoise, September 4--6, 2006The author studies exponential decay of the eigenfunctions of first-order (matrix) differential operators of the form $$ H = -i \sum_{j=1}^d A_j \frac{\partial}{\partial x_j} + V(x). $$ It is shown that under certain assumptions, the eigenfunctions obey estimates of the type $$ \int_{\Bbb R^d} |\psi(x)|^2 e^{2\delta x} \, dx < \infty. $$ The author emphasizes that these estimates are valid everywhere off the essential spectrum $\sigma_{\rm ess}$, not just below the minimum of $\sigma_{\rm ess}$