The Characterization Of Self-Adjoint Domains Of Two-Interval Differential Operators By Real-parameter Solutions And The Discreteness Of Spectrum

In this paper, we study the characterization of self-adjoint domains of two-interval differential operators and the discreteness of spectrum of several ordinary differential operators. Over the years, Sturm-Liouville problems with transmission conditions are concerned by many mathematical and physical researchers. While the problem with trans-mission conditions can be understood as special case of two-interval problem, i.e. with the coincide endpoint of two adjacent intervals whose left and right boundary conditions yield the transmission conditions. Partly motivated by this idea, in1986, Everitt-Zettl devel-oped a theory of self-adjoint realizations of Sturm-Liouville problems on two intervals in the direct sum of Hilbert spaces associated with these intervals. However the two-interval problem which are not merely the sum of self-adjoint operators from each of the separate intervals. It is interesting and important to observe that they involve interactions between the two intervals. Therefore in [9] the authors develop a "two-interval" theory. Since the spectrum of self-adjoint operators is real, the advantage of using real-parameter solutions to characterize self-adjoint domains is not only because it is, in general, easier to find explicit solutions but, more importantly, it yields information about the spectrum.In this paper, first of all, we give an complete characterization of all self-adjoint domains of differential operators on two intervals in the direct sum of Hilbert spaces in terms of real-parameter solutions. These two intervals with one endpoint of each interval is singular. A simple way of getting self-adjoint operators in a direct sum Hilbert space is to take the direct sum of self-adjoint operators from each of the separate Hilbert spaces. If these were all the self-adjoint operator realization from the two intervals there would be no need for a "two interval" theory. In fact, as noted in [9], there are many self-adjoint operators which are not merely the sum of self-adjoint operators from each of the separate intervals. These "new" self-adjoint operators involve interactions between the two intervals. These interactions may be'through'regular or singular endpoints. The regular self-adjoint interactions can be visualized as jumps of the solutions or their quasi- derivatives and the singular interactions can be described as jumps of Lagrange brackets involving the solutions.Then we also give an explicit characterization of all self-adjoint extensions of the two-interval minimal operator in terms of real-parameter solutions of the two intervals. These two intervals with all four endpoints are singular. These extensions yield'new' self-adjoint operators which are not merely direct sums of self-adjoint operators from the two intervals but involve interactions between the two intervals. These interactions are the interactions between singular endpoints. At singular interior points these interactions involve jump discontinuities of the Lagrange bracket of solutions. This result reduces to the case when one or two or three or four endpoints are regular.Furthermore, we study all self-adjoint two-interval realizations of the two equations of even order with real valued coefficients using Hilbert spaces but with the usual inner products replaced by appropriate multiples. we give an characterization of all self-adjoint extensions of the two-interval minimal operator with one endpoint of each interval is singular. The interplay of these multiples with the boundary conditions generates self-adjoint problems of even order with real coupling matrices K which are much more general than the coupling matrices from the'unmodified'theory.We also study the case when the four endpoints are singular in the modified direct sum Hilbert space with different inner product multiples. we give the characterization of all self-adjoint two-interval realizations of the two equations of even order with real valued coefficients. The self-adjoint boundary conditions obtained by us associated with real coupling matrices K are much more general. This result reduces to the case when one or two or three or four endpoints are regular.And we also study the eigenvalues of a regular fourth-order Sturm-Liouville problems depends on the problem. We observe that the eigenvalues of the regular fourth-order S-L problems depend not only continuously but smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter:an endpoint, a boundary condition, a coefficient, or the weight function, are found. Besides its theoretical importance, the continuous dependence of the eigenvalues and the eigenfunctions on the data is fundamental from the numerical point of view.Finally, we study the discreteness of spectrum of several ordinary differential opera-tors. First, we study the spectrum of a class of self-adjoint differential operators of even order. When the coefficients ak(x) of the differential operators are restricted by powers of ex, we give a sufficient condition on the coefficients to ensure that the spectrum of the differential operators is discrete; And we formulate necessary and sufficient conditions for the discreteness of the spectrum of the differential operators whose coefficients ak(x) may increase as powers of ex as x→∞. Second, the spectrum of self-adjoint differential operators of a class of symmetric differential expressions with exponential coefficients are considered. The last term coefficient which tends to infinity according to a certain way when the coefficients satisfying certain conditions. A sufficient condition is given which ensure that the spectrum is discrete. We further find that the discreteness of the spectrum of such differential operators not only determined by the last term coefficient which tends to infinity according to a certain way, Moreover, the middle term and the leading coeffi-cient which tends to infinity according to a certain way also can decide the discreteness of the spectrum. Finally, we give the range of essential spectrum of a class of self-adjoint differential operators with exponential coefficients.This thesis consists of eight chapters. In chapter1, we introduce the background about the problems what we study and the main results of this thesis; Chapter2is related symbols, concepts and properties involved in this thesis; Chapter3study the characterization of self-adjoint domains of differential operators on two intervals with one endpoint is singular of each interval; Chapter4study the characterization of self-adjoint domains of differential operators on two intervals with the endpoints all are singular; Chapter5study the characterization of self-adjoint domains of differential operators on two intervals with inner product multiples and one endpoint is singular of each interval; Chapter6study the characterization of self-adjoint domains of differential operators on two intervals with inner product multiples and the endpoints all are singular; Chapter7study the eigenvalues of a class of regular fourth-order Sturm-Liouville problems; Chapter8study the discreteness of the spectrum of several self-adjoint differential operators.