A ground state alternative for singular Schrödinger operators |
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Authors: | Yehuda Pinchover Kyril Tintarev |
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Institution: | a Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel b Department of Mathematics, Uppsala University, SE-751 06 Uppsala, Sweden |
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Abstract: | Let a be a quadratic form associated with a Schrödinger operator L=-∇·(A∇)+V on a domain Ω⊂Rd. If a is nonnegative on , then either there is W>0 such that for all , or there is a sequence and a function ?>0 satisfying L?=0 such that a?k]→0, ?k→? locally uniformly in Ω?{x0}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Ω. In the latter case, one has an inequality of Poincaré type: there exists W>0 such that for every satisfying there exists a constant C>0 such that for all . |
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Keywords: | primary 35J10 secondary 35J20 35J70 49R50 |
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