The essential self-adjointness of generalized Schrödinger operators (Q1058124)

The paper treats the essential self-adjointness of the generalized Schrödinger operator \[ A=-(1/2\rho)\sum^{n}_{i=1}D_ i(\rho D_ i) \] in the Hilbert space \(L^ 2(\Omega;\rho dx)\). In terms of related closed bilinear forms, a necessary and sufficient condition for the essential self-adjointness of \(A\) is given under general assumptions on \(\rho\) and for arbitrary domains \(\Omega\) in \({\mathbb{R}}^ n\). This criterion is used to prove that \(A\) is essentially self-adjoint in \(L^ 2({\mathbb{R}}^ n;\rho dx)\), if \(\rho\) is strictly positive and locally Lipschitz continuous on \({\mathbb{R}}^ n\). Furthermore, examples of non-essential self-adjointness and a complete discussion of the one dimensional case are included in the paper.