Non-real eigenvalues of a class of fourth order discontinuous differential operators with indefinite weight function

Abstract

The spectral theory of differential operators is one of the basic problems of differential operator theory. It has been an important tool in quantum mechanics and other fields. For the Sturm-Liouville Problems with weighted functions, the spectral theory of right-definite problems with self-adjoint boundary condition has been accomplished. But the spectral structure of indefinite problems, is quite different from and more complicated than that of right-definite problems. Compare with second order case, much less known for the fourth order indefinite differential operators. The present paper discusses a class of fourth order discontinuous differential operators. We obtained the estimate on the non-real eigenvalues, under separated boundary condition and the condition that weighted functions change signs for only once or any times.