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An Agmon–Allegretto–Piepenbrink principle for Schrödinger operators
被引：0
作者：
Stefano Buccheri
Luigi Orsina
Augusto C. Ponce
机构：
[1] University of Vienna,Faculty of Mathematics
[2] “Sapienza” Università di Roma,Dipartimento di Matematica
[3] Université catholique de Louvain,undefined
[4] Institut de Recherche en Mathématique et Physique,undefined
来源：
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
|
2022年
/
116卷
关键词：
Schrödinger operator;
AAP principle;
Poincaré inequality;
Singular potential;
Primary 35J10;
35R05;
46E35;
Secondary 35B05;
35J15;
35J20;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
We prove that each Borel function V:Ω→[-∞,+∞]\documentclass[12pt]{minimal}
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\begin{document}$$V : \Omega \rightarrow [{-\infty }, +\infty ]$$\end{document} defined on an open subset Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb R}^{N}$$\end{document} induces a decomposition Ω=S∪⋃iDi\documentclass[12pt]{minimal}
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\begin{document}$$\Omega = S \cup \bigcup _{i} D_{i}$$\end{document} such that every function in W01,2(Ω)∩L2(Ω;V+dx)\documentclass[12pt]{minimal}
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\begin{document}$$W^{1,2}_{0}(\Omega ) \cap L^{2}(\Omega ; V^{+} \,\mathrm {d}x)$$\end{document} is zero almost everywhere on S and existence of nonnegative supersolutions of -Δ+V\documentclass[12pt]{minimal}
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\begin{document}$$-\Delta + V$$\end{document} on each component Di\documentclass[12pt]{minimal}
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\begin{document}$$D_{i}$$\end{document} yields nonnegativity of the associated quadratic form ∫Di(|∇ξ|2+Vξ2).\documentclass[12pt]{minimal}
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\begin{document}$$ \int _{D_{i}} (|\nabla \xi |^2+V\xi ^2). $$\end{document}
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