The Spectral Function and Principal Eigenvalues for Schrödinger Operators

  title={The Spectral Function and Principal Eigenvalues for Schr{\"o}dinger Operators},
  author={Wolfgang Arendt and Charles J. K. Batty},
Let m 2 Lloc(R N ); 0 6= m+ in Kato’s class. We investigate the spectral function 7! s( + m)where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m is sufficiently large, we show that there exists a unique 1 > 0 such that s( + 1m) = 0. Under suitable conditions on m it follows that 0 is an eigenvalue of + 1m with positive eigenfunction. 

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