Uniqueness of Schrödinger operators restricted in a domain
The author considers an (open) domain \(D\) in \(\mathbb{R}^n\) with boundary \(\partial D\). Let \(C^\infty_0(D)\) be the space of all differentiable real functions on \(D\) with compact support. He considers the Schrödinger operator \(S:(-\Delta/2+ V, C^\infty_0(D))\), where \(V:D\to \mathbb{R}\) is a Borel measurable potential. The essential self-adjointness of \(S\) in \(L^2(D,dx)\) is equivalent to the uniqueness of the self-adjoint extension of \(S\) or the unique solvability of Schrödinger's equations in \(L^2(D)\), which has been studied by Kato, Reed and Simon and others. The main results and related works are studied in the first section. Some preliminaries and lemmas are given in section 2. The proofs of the lemmas and theorems are given in section 3, 4, 5. Some applications to the unique solvability of partial differential equations are presented in section 6. Finally, several interesting counterexamples and examples are presented in section 7.
scientific article; zbMATH DE number 1215686
scientific article