Eigenvalue counting estimates for a class of linear spectral pencils with applications to zero modes

Consider a linear spectral pencil of the form P ( − i ∇ ) − z Q ( x ) , z ∈ C . If P − 1 ∈ weak- L p and Q ∈ L p for some 1 < p < ∞ , it is shown that the total number of eigenvalues with | z | ⩽ R is bounded by C [ ‖ P − 1 ‖ L w p ⁎ ‖ Q ‖ L p R ] p . An application is made to estimate the frequency with which zero modes of the Weyl–Dirac operator occur when the magnetic potential is scaled.