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Zbl 1013.35021
Maz'ya, Vladimir G.; Verbitsky, Igor E.
(Verbitskij, Igor E.)
The Schrödinger operator on the energy space: Boundedness and compactness criteria. (English)
[J] Acta Math. 188, No.2, 263-302 (2002). ISSN 0001-5962; ISSN 1871-2509/e

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 $\Omega\subset \bbfR^d$ under mild restrictions on $\partial \Omega$. They obtain also criteria for the classical inequality $$\left |\int_{\bbfR^d} \bigl|u(x)\bigr |^2 V(x)dx\right |\le C_* \int_{ \bbfR^d} \bigl|\nabla u(x) \bigr|^2dx,\ u\in C_0^\infty (\bbfR^d),$$ to be hold, where the ``indefinite'' weight $V$ may change sign, or even be a complex-valued distribution on $\bbfR^d$, $d\ge 3$.
[Messoud Efendiev (Berlin)]

MSC 2000:: *35J10 Schroedinger operator
35B35 Stability of solutions of PDE
47F05 Partial differential operators
47H50 Potential operators
46N50 Appl. of functional analysis in quantum physics

Keywords: Schrödinger operator; energy space; boundedness and compactness criteria; Sobolev spaces

Cited in: Zbl 1161.31003 Zbl 1057.34104

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