On a domain characterization of Schrödinger operators with gradient magnetic vector potentials and singular potentials

Abstract:

Of concern are the minimal and maximal operators on ${L^2}({{\mathbf {R}}^n})$ associated with the differential expression \[ {\tau _Q} = \sum \limits _{j = 1}^n {(i\partial /\partial {x_j}} + {q_j}(x){)^2} + W(x)\] where $(q, \ldots ,{q_n}) = \operatorname {grad}Q$ for some real function $W$ on ${{\mathbf {R}}^n}$ and $W$ satisfies $c{\left | x \right |^{ - 2}} \leq W(x) \leq C{\left | x \right |^{ - 2}}$. In particular, for $Q = 0$, ${\tau _Q}$ reduces to the singular Schrödinger operator $- \Delta + W(x)$. Among other results, it is shown that the maximal operator (associated with the ${\tau _Q}$) is the closure of the minimal operator, and its domain is precisely \[ \operatorname {Dom}\left ( {\sum \limits _{j = 1}^n {{{(i\partial /\partial {x_j} + {q_j}(x))}^2}} } \right ) \cap \operatorname {Dom}(W),\] provided that $C \geq c > - n(n - 4)/4$.