The Spectrum Of Differential Operators With Indefinite Weight Function

The theory of ordinary differential operators is a systematic and comprehensive mathematical branch, which includes theories and methods in ordinary differential equations, functional analysis, operator algebra and function spaces. It is also a forceful tool in modern quantum mechanic, mathematics-physics equations and other technology fields. Some principal problems, including deficiency index, analysis of spectrum, eigenfunction expansion, inverse spectrum problem and description of self-adjoint domains etc. are studied in the theory of ordinary differential operators.In the paper, the spectrum of weighted differential operators, especially differential operators with indefinite weight function is investigated. It is known that if the weight function w>0, The problems become the familiar right-definite problems. In this case, L2(I,w) is a Hilbert space with the inner product (f,g)= fgw. This space is widely used to study the spectrum of the right-definite problems. If the weight function w changes sign, the problems become the right-indefinite ones. Compare with right-definite case, much less known for the indefinite problem. To study this problem three kinds of methods are used here: 1. introduce the spectral-curve and make use relationship between the left-definite problems and the right-definite problems, "transfer" results from the right-definite theory with weight function |w| to corresponding problems for the indefinite weight function w; 2. use the results and methods of operator theory in the Krein space L2(I,w) (indefinite inner product space) to investigate the spectrum of differential operators with indefinite weight function w ; 3. by the geometrical structure of self-adjoint boundary conditions toanalysis the dependence of left-definite problem on boundary conditions.The results we got include the following: 1. Using the basic nature of theoperators in Krein space and the base of the Sturm-Liouville operators, we construct a Krein space related to the differential operator. It is an important frame to investigate the indefinite differential operators. 2. we prove dependence of the left-definite sturm-Liouville problems on the boundary conditions and interval, and limits of the left-definite sturm-Liouville eigenvalues when interval shrinks to a point are given. 3. The spectrum of the regular four-order, in addition, high-order left-definite differential operators is studied, and some distribution natures are obtained. This is: its spectral are real eigenvalues and unbounded from below and from above, they can be indexed from negative infinity to positive infinity. 4. It is proved that if the leading coefficient in a four-order symmetrical differential expression is negative on a set with positive Lebesgue measure, then the minimal operator ( and hence any self-adjoint realization of the four-order symmetrical differential expression) is unbounded below. 5.For the singular 2n-order(n ≥ 2) left-definite differential operators, it is proved the existence of the eigenvalues below the essential spectrum. 6. Furthermore by the Minmax principle on the self-adjoint operators and spectral-curve, we consider the spectrum of the problem Ly = XRy which weight function is a self-adjoint operator.This paper contains six parts. Following the introduction, the spectrum theory in Krein space is given in section 2. In section 3 we discuss the dependence of left-definite Sturm-Liouville problems on the boundary conditions and the interval. The spectrums of the regular differential operators when weight function changes sign are given in section 4. In section 5 we studied the spectra of the singular four-order and high-order left-definite differential operator. The spectrum of the problem, which weight function is a self-adjoint operator, is achieved in section 6.