## Abstract

We prove the absence of positive eigenvalues of Schrödinger operators H = -Δ + V on Euclidean spaces ℝ^{n} for a certain class of rough potentials V. To describe our class of potentials fix an exponent q ∈ [n/2, ∞] (or q ∈ (1, ∞] if n = 2) and let β(q) = (2q - n)/(2g). For the potential V we assume that V ∈ L_{loc} ^{n/2}(ℝ^{n}) (or V ∈ L_{loc}^{r} (ℝ^{n}), r > 1, if n = 2) and lim_{R→∞} R^{β(q)}∥V∥_{Lq(R≤|x|≤2R)} = 0. Under these assumptions we prove that the operator H does not admit positive eigenvalues. The case q = ∞ was considered by Kato [K]. The absence of positive eigenvalues follows from a uniform Carleman inequality of the form ∥W _{m}u∥_{la(Lp′(q))(ℝn)} ≤ C _{q}∥W_{m}|x|^{β(q)} (Δ + 1)u∥ _{la(Lp(q))(ℝn)} for all smooth compactly supported functions u and a suitable sequence of weights W_{m}, where p(q) and p′(q) are dual exponents with the property that 1/p(q) - 1/p′(q) = 1/q.

Original language | English (US) |
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Pages (from-to) | 1029-1081 |

Number of pages | 53 |

Journal | Geometric and Functional Analysis |

Volume | 13 |

Issue number | 5 |

DOIs | |

State | Published - Dec 24 2003 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology