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Sturm-Liouville Problems Whose Leading Coefcient Function Changes Sign

Sturm-Liouville Problems Whose Leading Coefcient Function Changes Sign,Xifang Cao,Qingkai Kong,Hongyou Wu,Anton Zettl

Sturm-Liouville Problems Whose Leading Coefcient Function Changes Sign   (Citations: 6)
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For a given Sturm-Liouville equation whose leading coefcient function changes sign, we es- tablish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint bound- ary condition can be numbered in terms of the Prufer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also re- late this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme. In this paper, we study self-adjoint Sturm-Liouville problems (SLP's) associated with regular Sturm-Liouville equations (SLE's) (0.1) (p y0)0 + qy = wy on (a; b);
Published in 2003.
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    • ...We believe that this explicit representation of the selfadjoint BCs is an important preparation for an in-depth study of the dependence of the eigenvalues of a selfadjoint problem on its BC for the orders greater than or equal to 3 (see, e.g., [1, 3, 9, 12, 14 ]f or order 2), and we plan to undertake this task in forthcoming publications...
    • ...For a topological manifold M ,a topological M-bottle is a quotient space N that one obtains from M × [0, 1] via modeling M ×{ 0} by an equivalence relation on M to form a subset of N, to be called the top of N, and modeling M ×{ 1} by another equivalence relation on M to form a topological submanifold of N, to be called the bottom of N. In this case, M × (0, 1) is called the side of N. With the concept of topological ...

    Xifang Caoet al. Geometric aspects of high-order eigenvalue problems I. Structures on s...

    • ...In this case, our indexing scheme for eigenvalues of the associated regular problems is adopted from the recent paper of Binding and Volkmer [4] for separated BC’s and from [5] for coupled BC’s...
    • ...The singular problems with p > 0 were studied in [18], and the regular problems with p changing sign were investigated in [5]...
    • ...Consequently, the eigenvalues for coupled BC’s can be indexed according to the eigenvalue inequalities established in [5]...
    • ...Although the numbers of zeros of eigenfunctions for this case are dieren t from (i), they can be determined based on the sign changes of p on J, see [5] for details...
    • ...The following are analogues of Lemmas 4.1, 4.2 and Remark 4.1 for the case where p changes sign, see Theorem 3.10 in [5]...
    • ...Remark 4.2. For the case when p changes sign on J, it has been further proved in [5] that for n 2 N0, the n-th eigenvalue n = n(a0; b0; [AjB]) of SLP (4.1), (4.2) as a function of the endpoints a0; b0 2 (a; b) and the BC [AjB] 2 B is continuous whenever [AjB] 2 B n J ; and the continuous and discontinuous behavior of n(a0; b0; [AjB]) at a point where [AjB] 2 J is exactly the same as characterized in Lemma 4.6, no matter how a0; b0 change ...

    L. KONGet al. REGULAR APPROXIMATIONS OF SINGULAR STURM-LIOUVILLE PROBLEMS WITH LIMIT...

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