Bounds On Eigenvalues Of Several Differential Operators

With the extensive research on the spectral theory of differential operator,the spectral theory of right definite problem with self-adjoint boundary conditions has been accomplished.So,more and more researchers have paid attention to indefinite problems.As we all known,the indefinite problems may have non-real eigenvalues.Most researchers estimated the non-real eigenvalue of indefinite second order Sturm-Liouville Problems in different boundary conditions.But compare with second order case,much less known for the fourth order indefinite differential operators.On the other hand,the complex valued differential operators also may have non-real eigen-values,less researchers estimated the non-real eigenvalue of indefinite second order complex valued Sturm-Liouville Problems.Then,inspired by relative references,the paper will put the estimate of the non-real eigenvalues of second-order problems into fourth-order problems.Firstly,we investigate a class of fourth-order indefinite boundary value problem,where the boundary conditions are a kind of separated conditions,and the weighted function changes signs for only once or any times.The pure analysis and measure theory are used as tools to evaluate the upper bound of non-real eigenvalues.Secondly,we investigate a class of fourth-order regular indefinite boundary value problem,where the weight function gives rise to the existence of "ghost states".The upper bound of non-real eigenvalues about this problem is obtained.Then,the operator theories of Krein of space is used for the estimate of the lower bound of non-real eigenvalues.At the same time,with the oscillation of weighted function,the more accurate evaluation on the lower bound of non-real eigenvalue is given.Finally,we investigate a class of fourth-order complex valued problem.Under some condition,we have obtained the lower bound of the real parts of eigenvalues and the upper bound of the imaginary part of eigenvalue.