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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 \(\Omega\subset \mathbb{R}^d\) under mild restrictions on \(\partial \Omega\). They obtain also criteria for the classical inequality \[ \left |\int_{\mathbb{R}^d} \bigl|u(x)\bigr |^2 V(x)dx\right |\leq C_* \int_{ \mathbb{R}^d} \bigl|\nabla u(x) \bigr|^2dx,\;u\in C_0^\infty (\mathbb{R}^d), \] to be hold, where the “indefinite” weight \(V\) may change sign, or even be a complex-valued distribution on \(\mathbb{R}^d\), \(d\geq 3\).

MSC:
35J10 Schrödinger operator, Schrödinger equation
35B35 Stability in context of PDEs
47F05 General theory of partial differential operators
47H50 Potential operators (MSC2000)
46N50 Applications of functional analysis in quantum physics
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