On the essential selfadjointness of a nonsemibounded Schrödinger operator with strong singularities in the potential
The adjointness of the Schrödinger operator in the setting of \(L_2(G)\), where \(G\) is an open subset of \(\mathbb{R}^n\) but \(G\neq\mathbb{R}^n\), has been studied by several authors. A discussion of this problem can be found in the monograph of \textit{M. Reed} and \textit{B. Simon} [ Methods of modern mathematical physics. II: Fourier analysis, selfadjointness. New York etc.: Academic Press (1975; Zbl 0308.47002)]. The author offers in this short communication (without proofs) three theorems, which can be found in the literature, but he offers much weaker hypothesis. He weakens the smoothness conditions imposed on the potential and omits the assumption that the potential is semibounded. The Schrödinger operator discussed is of the form \(Su=-(\nabla- ib(x)) u+ q(x)u\), where \(b(x)\) is a \(C^1(G)\) \(n\)-dimensional vector with real components and \(q(x)\) is a real function. The main theorem is Theorem 3, which asserts the essential selfadjointness of the related Hamiltonian for a system of particles with different masses, subjected to the Coulomb attraction. A similar result was proved by the author [ Mat. Fiz. Anal. Geom. 5, 149--165 (1998; Zbl 0951.35034)]{} but under more restrictive conditions:
scientific article; zbMATH DE number 1916203