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

## TL;DR

This paper rigorously defines Feynman path integrals for magnetic Schrödinger equations with polynomially growing electromagnetic potentials, expanding the mathematical framework to include more physically relevant unbounded potentials.

## Contribution

It establishes a rigorous mathematical definition of Feynman path integrals for magnetic Schrödinger equations with polynomially growing potentials, including unbounded electromagnetic potentials.

## Key findings

- Path integrals are well-defined as $L^2$-valued continuous functions in time.
- Includes potentials with polynomial growth in spatial variables.
- Extends previous definitions to unbounded electromagnetic potentials.

## 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.