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The Schrödinger operator on the energy space: Boundedness and compactness criteria. (English) Zbl 1013.35021

This paper deals with the property of the Schrödinger operator on the energy space. The authors present an complete solution to the problem of the relative form-boundedness of the potential energy operator $V$ with respect to the Laplacian $-{\Delta }$, which is fundamental to quantum mechanics. Moreover, the authors give both boundedness and compactness criteria for Sobolev spaces on domains ${\Omega }\subset {ℝ}^{d}$ under mild restrictions on $\partial {\Omega }$. They obtain also criteria for the classical inequality

$\left|{\int }_{{ℝ}^{d}}{\left|u\left(x\right)\right|}^{2}V\left(x\right)dx\right|\le {C}_{*}{\int }_{{ℝ}^{d}}{\left|\nabla u\left(x\right)\right|}^{2}dx,\phantom{\rule{4pt}{0ex}}u\in {C}_{0}^{\infty }\left({ℝ}^{d}\right),$

to be hold, where the “indefinite” weight $V$ may change sign, or even be a complex-valued distribution on ${ℝ}^{d}$, $d\ge 3$.

##### MSC:
 35J10 Schrödinger operator 35B35 Stability of solutions of PDE 47F05 Partial differential operators 47H50 Potential operators (MSC2000) 46N50 Applications of functional analysis in quantum physics
##### References:
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