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A ground state alternative for singular Schrödinger operators
| Content Provider | Semantic Scholar |
|---|---|
| Author | Pinchover, Yehuda |
| Copyright Year | 2008 |
| Abstract | Let a be a quadratic form associated with a Schrödinger operator L = −∇ · (A∇) + V on a domain Ω ⊂ R. If a is nonnegative on C∞ 0 (Ω), then either there is W > 0 such that ∫ W |u|2 dx ≤ a[u] for all C∞ 0 (Ω;R), or there is a sequence φk ∈ C ∞ 0 (Ω) and a function φ > 0 satisfying Lφ = 0 such that a[φk] → 0, φk → φ 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 ψ ∈ C∞ 0 (Ω;R) satisfying ∫ ψφdx 6= 0 there exists a constant C > 0 such that C−1 ∫ W |u|2 dx ≤ a[u] +C ∣∣∫ uψ dx ∣∣2 for all u ∈ C∞ 0 (Ω;R). 2000 Mathematics Subject Classification. Primary 35J10; Secondary 35J20, 35J70, 49R50. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0411658v1.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Ground state Hamiltonian (quantum mechanics) Mathematics Subject Classification Neoplasm Metastasis Schrödinger Singular Social inequality |
| Content Type | Text |
| Resource Type | Article |
