Remarks on essential self-adjointness of Schrödinger operators with singular electrostatic potentials.

*(English)*Zbl 0776.35009For Schrödinger operators \(Tu=-\Delta u+Vu\) on \(D(T):=C_ 0^ \infty(\mathbb{R}^ N)\) \((N\geq 3)\) essential self-adjointness is proved under the assumption \(V=V_ 1+V_ 2\) with real valued \(V_ i\in L^ 2_{\text{loc}}(\mathbb{R}^ N)\), where \(V_ 1(x)\geq-c| x|^ 2\) with some \(0\leq c\in\mathbb{R}\) and \(0\geq V_ 2\in K^ N\) where \(K^ N\) denotes the so-called Kato-class. Based on his meanwhile famous distributional inequality T. Kato [Isr. J. Math. 13, 135-148 (1972; Zbl 0246.35025)] proved essential self-adjointness of \(T\) assuming an additional growth condition for \(V_ 2\). The last restriction can be omitted as was shown subsequently by several authors using completely different methods [see e.g. R. Jensen, Commun. Partial Differ. Equations 3, 1053-1076 (1978; Zbl 0391.35022) or B. Simon, Bull. Am. Math. Soc., New Ser. 7, 447-526 (1982; Zbl 0524.35002)].

The proof presented now is quasi a corollary of the author’s recent results [Math. Z. 203, No. 1, 129-152 (1990; Zbl 0697.35017)] combined with his some earlier considerations [Math. Z. 138, 53-70 (1974; Zbl 0317.35028)] and turns out to be completely elementary. In addition locally uniformly \(L^ \infty\)-estimates and decay results are given.

The proof presented now is quasi a corollary of the author’s recent results [Math. Z. 203, No. 1, 129-152 (1990; Zbl 0697.35017)] combined with his some earlier considerations [Math. Z. 138, 53-70 (1974; Zbl 0317.35028)] and turns out to be completely elementary. In addition locally uniformly \(L^ \infty\)-estimates and decay results are given.

Reviewer: C.G.Simader (Bayreuth)

##### MSC:

35J10 | Schrödinger operator, Schrödinger equation |

47F05 | General theory of partial differential operators |

35P05 | General topics in linear spectral theory for PDEs |