, Volume 7, Issue 1, pp 415-436

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

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Abstract

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.