# zbMATH — the first resource for mathematics

Remarks on essential self-adjointness of Schrödinger operators with singular electrostatic potentials. (English) Zbl 0776.35009
For 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.

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 47F05 General theory of partial differential operators 35P05 General topics in linear spectral theory for PDEs
Full Text: