A Feynman–Kac type formula for relativistic Schrödinger operators with unbounded vector potential and spin 1=2 is given in terms of a three-component process consisting of a Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic elds and magnetic elds with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound states is established separately for growing and for decaying potentials by using martingale methods.
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Fumio Hiroshima, Takashi Ichinose, József Lőrinczi, Probabilistic Representation and Fall-Off of Bound States of Relativistic Schrödinger Operators with Spin 1/2. Publ. Res. Inst. Math. Sci. 49 (2013), no. 1, pp. 189–214DOI 10.4171/PRIMS/102