Holomorphic spectral geometry of magnetic Schrödinger operators on Riemann surfaces

Abstract

We study magnetic Schrödinger operators on line bundles over Riemann surfaces endowed with metrics of constant curvature. We show that for harmonic magnetic fields the spectral geometry of these operators is completely determined by the Bochner Laplacians of the line bundles. Therefore we are led to examine the spectral problem for the Bochner Laplacian ∇ ∗ ∇ of a Hermitian line bundle L with connection ∇ over a Riemann surface S. This spectral problem is analyzed in terms of the natural holomorphic structure on L defined by the Cauchy–Riemann operator associated with ∇. By means of an elliptic chain of line bundles obtained by twisting L with the powers of the canonical bundle we prove that there exists a certain subset of the spectrum σ hol ( ∇ ∗ ∇ ) such that the eigensections associated with λ ∈ σ hol ( ∇ ∗ ∇ ) are given by the holomorphic sections of a certain line bundle of the elliptic chain. For genus p = 0 , 1 we prove that σ hol ( ∇ ∗ ∇ ) is the whole spectrum, whereas for genus p > 1 we get a finite number of eigenvalues.