Potential Analysis

, Volume 7, Issue 1, pp 415–436

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

  • Wolfgang Arendt
  • Charles J. K. Batty

DOI: 10.1023/A:1017928532615

Cite this article as:
Arendt, W. & Batty, C.J.K. Potential Analysis (1997) 7: 415. doi:10.1023/A:1017928532615


Let m ∈ \(L_{loc}^1 (\mathbb{R}^N ) \), 0 ≠ m+ in Kato's class. We investigate the spectral function λ ↦ s(Δ + λm) where s(Δ + λm) denotes the upper bound of the spectrum of the Schrödinger operator Δ + λm. In particular, we determine its derivative at 0. If m- is sufficiently large, we show that there exists a unique λ1 > 0 such that s(Δ + λ1m) = 0. Under suitable conditions on m+ it follows that 0 is an eigenvalue of Δ + λ1m with positive eigenfunction.

Principal eigenvalueSchrödinger semigroupexponential stabilityspectral boundBrownian motion.

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Wolfgang Arendt
  • Charles J. K. Batty

There are no affiliations available