Lower Bounds for the Spectra of Schrِdinger Operators with Magnetic Fields

An explicit representation of lower bounds for the spectra of Schrödinger operators with magnetic fields on σ-compact Riemannian manifolds is given, using the positivity of the Pauli Hamiltonian. This representation is applied to show some asymptotic properties of a stochastic oscillatory integral and the transverse analyticity of the law of a stochastic line integral.