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Boundary problems of the second order with an indefinite weight-function. (English) Zbl 0668.35073

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

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
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