The Number Of Real-Parameter Square-Integrable Solutions And The Qualitative Analysis Of The Spectrum

In this paper, we investigate the relationship between the number of real-parameter square-integrable solutions of the differential equations and the qualitative analysis of the spectrum of the differential operators.We notice that, since the spectrum of self-adjoint operators is real there is a close connections between the spectral analysis of the self-adjoint operator and the null space of the real-parameter square-integrable solu-tions. At the same time, we notice that the number of the real-parameter square-integrable solutions is determined by the coefficients of differen-tial expression, the essential spectrum of the differential operators and the deficiency indices are also only related to the coefficients of differen-tial expression. There should be close connection among the deficiency indices, the number of the real-parameter square-integrable solutions and the essential spectrum of the differential operators. Investigating the re-lationship among them is a important problem. We adopt new measures to qualitatively study the distribution of spectrum in terms of the num-ber of real-parameter square-integrable solutions of the singular differential operators.For differential operators with one singular endpoint, in 1987, Weid-mann in his book [86] raised a famous conjecture:"If for everyλ∈Ⅰ(?)R, there exist'sufficient many'square-integrable solutions of the differential operators, thenⅠcontains no points of the essential spectrum." We should be noticed that, in 1996, Remling [62] has shown that, for the case of n= 2, d= 1, r(λ)= d= 1 for allλin some open intervalⅠ, it does not imply that there is no essential spectrum inⅠ. Under what conditions the Weidmann's conjecture holds, i.e., what additional condition guarantees no essential spectrum inⅠ?We investigate this problem, and give a new characterization of self-adjoint domains. Firstly, we analyze the relationship between the separated boundary conditions and other boundary conditions. On this basis, using the approaching of regular operator in the sense of strong resolvent con-vergence, we prove that, assume there exists an open intervalⅠof the real line such that the differential equation has d linearly independent square-integrable solutions for everyλ∈Ⅰ, then for any self-adjoint realization, the intersection of its continuous spectrum andⅠis empty. Secondly, we give an assumption about the analytic dependence of solutions of the differ-ential equations on the parameterλand prove that if the above assumption holds, then there is no essential spectrum inⅠ, i.e.. the spectrum is discrete inⅠ. Our results give an complete answer to Weidmann's conjecture [86]: If there exist'sufficient many'real-parameter square-integrable solutions of the differential equations in some open interval, add the assumption about the analytic dependence of the solutions on the parameter A, there is no essential spectrum in the open interval. Next we investigate the relationship between the number of real-parameter square-integrable solutions of the differential equations with two singular endpoints and the distribution of the spectrum. We give the com-plete characterization of self-adjoint domains by the real-parameter square-integrable solutions. Firstly we give a new decomposition of the maximal domain. The key point is separating the boundary conditions of the two singular endpoints, connecting real-parameter square-integrable solutions at the two endpoints in terms of the functions in the maximal domain. It is this representation of the maximal domain which leads to a uniformly method to solve the problem, i.e., as a special case we obtain the repre-sentation of the maximal domain when one or both endpoints are regular just regard the deficiency index at the regular endpoint as n. Then we give self-adjoint boundary conditions and the complete characterization of self-adjoint domain in terms of the solutions of the differential equation.Further, we give the relationship between the number of real-parameter square-integrable solutions of the differential operators with two singular endpoints and distribution of the spectrum. Firstly, we prove that for the two singular endpoints differential operators, the number of the real-parameter square-integrable solutions is possible less than d, equal to d and greater than d (This is impossible for one singular endpoint case). This implies that there is a essentially difference between the two singular end-points case and one singular endpoint case when we study the qualitative analysis of spectrum, and two singular endpoints case is not a simple ex- tension of one singular endpoint case. We give a concise proof about the open problem raised by Weidmann (see [86]), i.e., we prove that if the num-ber of the real-parameter square-integrable solutions r(λ)< d, then A is a point in essential spectrum. And we prove that if r(λ)= d. for everyλ∈Ⅰand the assumption about the analytic dependence of the solutions on the parameterλholds onⅠ, then eigenvalues of every self-adjoint realization have no accumulation point inⅠ. We prove that if r(λ)> d, thenλis an eigenvalue for self-adjoint operator realizations, it implies that when the number of the real-parameter square-integrable solutions is "too many' there may be essential spectrum inⅠ. This result is completely different from the one singular endpoint case.The canonical forms of the self-adjoint boundary conditions of the differential equations is the basis of investigating the effects of the boundary conditions of differential operators on the distribution of eigenvalues of differential operators. We give the canonical forms of four order differential operators for d = 4 (It contains the cases of two regular endpoints and two singular endpoints), d= 3 and d = 2. As there are so many canonical forms under the four order case, we find a uniform way to obtain the canonical forms, i.e., we give a "fundamental canonical form", other canonical forms can be obtained by the "fundamental canonical form"...