The number of real-parameter solutions and the spectrum of differential operators
Speaker:Prof. Jiong Sun
Department of Mathematics
Inner Mongolia University
Date & Time:5 Mar 2012 (Monday) 10:30
Venue:J320
Organized by:Department of Mathematics

Abstract

The spectrum of a self-adjoint ordinary differential operator in Hilbert space H = L2(a,b) is real, consists of eigenvalues of finite multiplicity, of essential and of continuous spectrum. A number λ is an eigenvalue if the corresponding differential equation has a nontrivial solution which satisfies the boundary conditions. On the other hand, the essential spectrum is independent of the boundary conditions and thus depends only on the coefficients of the equation. This dependence is implicit and highly complicated. We give a comprehensive account of the relationship between the square-integrable solutions for real values of the spectral parameter λ and the spectrum of self-adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d. We show that if, for all real parameter λ (μ1,μ2) there are d linearly independent square-integrable solutions, then there is no continuous spectrum in (μ1,μ2 ); and if the initial values of solutions analytically depend on the real-parameters, then the essential spectrum of differential operators generated by the differential expression is empty in this open interval, it means the spectrum is discrete. The proof is based on a new characterization of self-adjoint domains and on limit-point (LP) and limit-circle (LC) solutions.

Biography

Professor Jiong Sun graduated from the Department of Mathematics at Inner Mongolia University, who is also presently the Professor of the Department. He received various awards, such as, National distinguished university teacher award, National outstanding scientific and technological worker, Natural Science Award of Inner Mongolia Autonomous Region (Third prize), National teaching achievement award (Second Prize). Currently, he is working on "On the discreteness of spectrum of differential operators, National Nature Science". His publications include, On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices, Extension of functions on weighted Sobolev spaces, and entropy numbers of Sobolev embeddings domains with finite measure, Embedding theorems and the spectra of certain differential operators etc.