Non-relativistic limit of generalized relativistic Pauli operators by Feynman-Kac formulae

Abstract

The non-relativistic limit of a generalized relativistic Pauli operator\[H_c^{S,\alpha}=\left(2c^{\beta}\bigl(\sigma\cdot(-i\nabla-a)\bigr)^2+(mc^\gamma)^{2/\alpha}\right)^{\alpha/2}-mc^\gamma+V\]on $L^2(\mathbb{R}^3;\mathbb{C}^2)$ is investigated under the constraint$2\alpha=\gamma\beta+\gamma^2$.This operator generalizes the relativistic Pauli operator within the framework of Bernstein functions.The associated heat semigroup $e^{-tH_c^{S,\alpha}}$ admits a Feynman--Kac representation involving Brownian motion, a subordinator, and a Poisson process.Using this representation, we prove that the semigroup $e^{-tH_c^{S,\alpha}}$ converges strongly to $e^{-tH^{S,\alpha}}$ as $c\to\infty$, where the limiting generator is given by\[H^{S,\alpha}=\frac{\alpha}{2m^{\frac{2}{\alpha}-1}}\bigl(\sigma\cdot(-i\nabla-a)\bigr)^2+V.\]The non-relativistic limit of a generalized relativistic Schr\"odinger operator is also investigated.