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

The non-relativistic limit of a generalized relativistic Pauli operator$$H_c^{S,\alpha }={\left(2c^{\beta }\right(\sigma \cdot (-i\mathrm{\nabla }-a){)}^2+(m{c}^{\gamma }{)}^{2\mathrm{/}\alpha })}^{\alpha \mathrm{/}2}-m{c}^{\gamma }+V$$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 $e^{-t{H}_c^{S,\alpha }}$ admits a Feynman–Kac representation involving Brownian motion, a subordinator, and a Poisson process.Using this representation, we prove that the semigroup $e^{-t{H}_c^{S,\alpha }}$ converges strongly to $e^{-t{H}^{S,\alpha }}$ as $c\to \mathrm{\infty }$, where the limiting generator is given by$$H^{S,\alpha }=\frac{\alpha }{2m^{\frac{2}{\alpha }-1}}(\sigma \cdot (-i\mathrm{\nabla }-a){)}^2+V\mathrm{.}$$The non-relativistic limit of a generalized relativistic Schrödinger operator is also investigated.