On the essential selfadjointness of stochastic Schrödinger operators
Let \(V_{\omega}\) be a random field on \({\mathbb{R}}^ d\) and consider the random Schrödinger operator \(H_{\omega}=-\Delta +V_{\omega}\). Let furthermore \(C_ y\) be the unit cube around the point y. The main result of the paper states that \(H_{\omega}\) is essentially selfadjoint on \(C_ 0^{\infty}({\mathbb{R}}^ d)\) if \(E(|(\int_{C_ y}| V_{\omega}(x)|^ qdx)^{1/q}|^ r)\leq C<\infty\) for \(q>q_ 0(d)\) and \(r>r_ 0(d,q)\) (E denotes the expectation with respect to \(\omega)\). Moreover it is clear from the proof that adding a potential \(W(x)\geq -cx^ 2\)-d does not disturb the essential selfadjointness \(q_ 0(d)=\max(2,d/2).\) There is some confusion about \(r_ 0(d,q)\) in the paper under review. The correct \(r_ 0\) is given by \(r_ 0(d,q)=dq/2(q-p)\) which should replace (3) in the paper. In the last section applications are discussed which cover many random potentials of physical and mathematical interest.
scientific article; zbMATH DE number 3864234