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An improved Hardy-Sobolev inequality in \(W^{1,p}\) and its application to Schrödinger operators. (English) Zbl 1089.35035
Summary: We prove new Hardy-like inequalities with optimal potential singularities for functions in \(W^{1,p}(\Omega)\), where \(\Omega\) is either bounded or the whole space \(\mathbb R^n\) and also some extensions to arbitrary Riemannian manifolds. We also study the spectrum of perturbed Schrödinger operators for perturbations which are just below the optimality threshold for the corresponding Hardy inequality.

MSC:
35P05 General topics in linear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35J10 Schrödinger operator, Schrödinger equation
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
26D15 Inequalities for sums, series and integrals
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