## Abstract

We prove that the resonance counting functions for Schr̈odinger operators H_{V} = -δ+V on L^{2}(ℝ^{d}), for d ≥ 2 even, with generic, compactly-supported, real- or complex-valued potentials V, have the maximal order of growth d on each sheet Λ_{m}, m ∈ Znf0g, of the logarithmic Riemann surface. We obtain this result by constructing, for eachm ∈ ℤ\{0}, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet Λ_{o}. determine the poles on Λ _{m}. We prove that the order of growth of the counting function is related to a suitable estimate on this function that we establish for generic potentials. We also show that for a potential that is the characteristic function of a ball, the resonance counting function is bounded below by Cmr ^{d} on each sheet Λ_{m}, m ∈ ℤ\{0}.

Original language | English |
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Pages (from-to) | 621-660 |

Number of pages | 40 |

Journal | Indiana University Mathematics Journal |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - 2010 |

## Keywords

- Counting function
- Resonances
- Schrödinger operators

## ASJC Scopus subject areas

- Mathematics (all)