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

Elton, Daniel and Ta, Tri Ngoc (2012) Eigenvalue counting estimates for a class of linear spectral pencils with applications to zero modes. Journal of Mathematical Analysis and Applications, 391 (2). pp. 613-618. ISSN 0022-247X

Abstract

Consider a linear spectral pencil of the form P(−i∇)− zQ(x), z ∈ C. If P−1 ∈ weak-Lp and Q ∈ Lp for some 1 < p <∞, it is shown that the total number of eigenvalues with |z|<R is bounded by C[||P−1||∗Lpw||Q||Lp 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.