Schr\"odinger operators with singular potentials and magnetic fields

V. N. Kolokoltsov

Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 897-917

Résumé

A formal Schrödinger operator of the form $$ H=\biggl(-i\frac\partial{\partial x}+A(x)\biggr)^2+V(x), $$ in ${\mathbb R}^d$ is considered, where $A$ is a bounded measurable vector-valued function and both $V(x)$ and $\operatorname{div}A$ are measures satisfying certain additional conditions. It is shown that one can give meaning to such an operator as a lower bounded self-adjoint operator in $L^2({\mathbb R}^d)$. The corresponding heat kernel is constructed and its small-time asymptotics are obtained. A rigorous Feynman path integral representation for the solutions of the heat and Schrödinger's equations with generator $H$ is given.