On the Feynman path integral for the magnetic Schroedinger equation with a polynomially growing electromagnetic potential

Abstract

The Feynman path integrals for the magnetic Schroedinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For example, we can handle electromagnetic potentials $(V,A_{1},A_{2},...,A_{d})$ such that $V(t,x) = |x|^{2(l+1)} + $`` a polynomial of degree $(2l + 1)$ in $x$ " ($l = 0,1,2,...$) and $A_{j}(t,x)$ are polynomials of degree $l$ in $x$. The Feynman path integrals are defined as $L^2$-valued continuous functions with respect to the time variable.