Counting function of magnetic eigenvalues for non-definite sign perturbations

We consider the perturbed operator $H(b,V):=H(b,0)+V$, where $H(b,0)$ is the $3$d Hamiltonian of Pauli with non-constant magnetic field, and $V$ is $\mathit{\text{a}}\mathit{\text{non-definite}}\mathit{\text{sign}}\mathit{\text{electric}}\mathit{\text{potential}}$ decaying exponentially with respect to the variable along the magnetic field. We prove that the only resonances of $H(b,V)$ near the low ground energy zero of $H(b,0)$ are its eigenvalues and are concentrated in the semi axis $(-\mathrm{\infty },0)$. Further, we establish new asymptotic expansions, upper and lower bounds on their number near zero.