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

Research output: Contribution to journalJournal article

Published

Journal publication date15/07/2012
JournalJournal of Mathematical Analysis and Applications
Journal number2
Volume391
Number of pages6
Pages613-618
Original languageEnglish

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.