Eigenvalue counting estimates for a class of linear spectral pencils with applications to zero modes
Let \(P, Q\) be functions from \({\mathbb R}^d\) to the space of \(m\times m\) matrices. The main assumption is that \(Q\) is an \(L^p\) function (using the matrix operator norm for its range) for some \(1<p<\infty\), while the pointwise inverse of \(P\) is a weak-\(L^p\) function. The authors consider the pencil of operators \(P(-i \nabla)-zQ(x)\), \(z\in {\mathbb C}\), on \(L^2({\mathbb R}^d,{\mathbb C}^m)\), where \(Q(x)\) is a multiplication operator and \(P(-i \nabla)\) is a multiplication operator on the Fourier space. It is shown that the total number of eigenvalues with \(|z|\leqslant R\) is bounded by \(C[||P^{-1}||^\ast_{L^P_w}||Q||_{L^p}R]^p\). A motivating example for this study is the Weyl-Dirac operator, a corollary of the main theorem estimates the frequency with which zero modes occur when the magnetic potential is scaled.
scientific article; zbMATH DE number 6030456
scientific article