Allegretto, W.; Mingarelli, A. B. Boundary problems of the second order with an indefinite weight-function. (English) Zbl 0668.35073 J. Reine Angew. Math. 398, 1-24 (1989). We study the behavior of the spectrum of boundary problems of the form \(Au=\lambda Bu\) where A is a symmetric second order ordinary/partial differential operator with some negative spectrum and B is a multiplication operator by a generally (sign) indefinite weight-function. The first part of the paper considers the case where A is uniformly elliptic. In this case we obtain, in particular, min.max. estimates for some of the real eigenvalues \(\lambda\) and existence criteria for both real and complex \(\lambda\). Extensive use is made of ideas from perturbation theory and from the theory of two parameter problems. The later parts of the paper deal with similar questions in the case \(n=1\) but under more general conditions on the coefficients, so that A need not be uniformly elliptic, and indeed may not even be elliptic. Estimates are then obtained, in particular, for the Haupt and Richardson indices. Reviewer: W.Allegretto Cited in 13 Documents MSC: 35P15 Estimates of eigenvalues in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 34L99 Ordinary differential operators Keywords:behavior of the spectrum; symmetric; negative spectrum; multiplication operator; indefinite weight-function; uniformly elliptic; min.max. estimates; existence; perturbation theory; Haupt and Richardson indices PDFBibTeX XMLCite \textit{W. Allegretto} and \textit{A. B. Mingarelli}, J. Reine Angew. Math. 398, 1--24 (1989; Zbl 0668.35073) Full Text: Crelle EuDML