The Dependence Of The Eigenvalues On The Potential Function And Weight Function Of The Left Definite Sturm-Liouville Problem

In recent years,the issue of Sturm-Liouville(S-L)has become a hot topic among scholars.The study of the dependence of eigenvalues on potential functions and weight functions has attracted many scholars attention.That is the S-L inverse problem.On the basis of previous research,this paper studies the problem of left definite S-L.First the monotonicity of interval endpoints is proved by comparing theorems and eigenvalues:The eigenvalues in the [0,1]interval tend to be one of the eigenvalues in the partial interval when the potential function on some intervals tends to infinity.Secondly,the Liouville theorem of the entire function is applied and the asymptotic estimate of the fundamental solution of S-L equation when the spectral parameter tends to infinity,It is concluded that the infinite set spectral information can uniquely determine the potential function in the whole interval.Specific research contents are as follows:In the first chapter,the research development status of the S-L theory and the work of this paper are introduced.In the second chapter,the S-L equation of the weight function is divided into piecewise function,given that the potential function on the negative interval is known,the sufficient condition of the entire potential function can be uniquely determined.In the third chapter,dependence of eigenvalues of S-L problem whose weight function ?(?)1 the potential function.By the comparison theorem,it proves that when the potential function in the part interval tends to infinity,the eigenvalues in the interval asymptotically approached some eigenvalue of the part interval.In the fourth chapter,inverse problem of partial spectral information of S-L operator.Given the potential function on a partial interval,the infinite set of spectral information can uniquely determine the potential function on the entire interval.