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Strongly singular Sturm-Liouville problems. (English) Zbl 0994.34013
In the usual Sturm-Liouville theory, in order to use the spectral theorem for selfadjoint operators in Hilbert spaces, appropriate boundary conditions are used together with the Sturm-Liouville equation $-(ru'(x))'+ p(x) u(x)=\lambda m(x) u(x)\quad\text{on }(a,b)$ to form selfadjoint eigenvalue problems. For example, for the Fourier equation $$-u''=\lambda u$$ on $$(0,+\infty)$$, such a boundary condition has the form $$u(0)\cos\alpha+ u'(0)\sin\alpha= 0$$ with $$\alpha\in [0,\pi)$$, since the endpoint $$+\infty$$ is of the limit-point type.
Thus, for a singular Sturm-Liouville equation, i.e., when one of $$1/r$$, $$p$$ and $$m$$ is not Lebesgue integrable on $$(a,b)$$, some very “natural” eigenvalue problems are nonselfadjoint. For example, the problem consists of the Fourier equation on $$(0,+\infty)$$ and the periodic boundary condition, i.e., $$u(0)= u(+\infty)$$ and $$u'(0)= u'(+\infty)$$, is nonselfadjoint. The eigenvalues of such problems are complicated in general. For example, $$0$$ is the only eigenvalue of the above concrete problem. The author deals with this type of eigenvalue problems.
Here, $$r: (a,b)\to \mathbb{R}^+$$ is continuous, $$p,m\in L^1_{\text{loc}}((a,b),\mathbb{R})$$, $$p\geq 0$$, and there is a $$c\in (a,b)$$ such that $$1/r\in L^1(a,c)$$, $$1/r\not\in L^1(c,b)$$, $$pr_1\not\in L^1(a,c)$$, and $$pr_1\not\in L^1(c,b)$$, where $$r_1(t)= \int^t_a ds/r(s)$$. The eigenvalue problem considered here consists of a Sturm-Liouville equation satisfying these assumptions and the Dirichlet boundary condition $$u(a)= u(b)= 0$$, and is said to be strongly singular.
The main results concerning the eigenvalues can be stated as follows: If $$m\in L^1(a,c)$$ and $$mr_1\in L^1(c,b)$$ for some $$c\in (a,b)$$, then all eigenvalues are simple, and together with $$0$$ they can be ordered to form a sequence $\cdots<\lambda_{-2}< \lambda_{-1}< 0< \lambda_1< \lambda_2<\cdots\;.$ Each of the sequences $$\dots,\lambda_{-2},\lambda_{-1}$$ and $$\lambda_1,\lambda_2,\dots$$ can be void, finite, or infinite with infinity as its limit. If, in addition, $$mr^{3/2}_1\in L^1(c,b)$$ for some $$c\in (a,b)$$ and $$\{t\in (a,b); m(t)> 0\}$$ has a positive measure, then $$\lambda_1$$ exists and is given by $\min\left\{ \int^b_a (ru^{\prime 2}+ pu^2)\Biggl/ \int^b_a mu^2;\;\begin{matrix} u\in C[a,b),\;u(a)= 0\\ \sqrt pu,\sqrt ru'\in L^2(a,b), \int^b_a mu^2> 0\end{matrix}\right\}.$ A maximum principle and an anti-maximum principle concerning $$\lambda_1$$ are proved. Moreover, the author also gives examples to show that under the assumption in this paper, all four possibilities of the limit-point and limit-circle classification occur.

##### MSC:
 34B24 Sturm-Liouville theory 34B27 Green’s functions for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations
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##### References:
 [1] Amann, SIAM Rev. 18 pp 620– (1976) [2] : A Topological Introduction to Nonlinear Analysis, Birkhauser, Boston, 1993 [3] Cecchi, J. Diff. Equations 82 pp 15– (1989) [4] Cecchi, J. Diff. Equations 99 pp 381– (1992) [5] Cecchi, J. Diff. Equations 118 pp 403– (1995) [6] Duhoux, Proc. Roy. Soc. Edinburgh 128A pp 525– (1998) · Zbl 0983.34015 · doi:10.1017/S0308210500021636 [7] Duhoux, Nonlinear Anal. 38 pp 897– (1999) [8] Duhoux, Tatra Mt. Math. Publ. 19 pp 1– (2000) [9] Hajmirzaahmad, SIAM Rev. 34 pp 614– (1992) [10] and : Singular Self-Adjoint Sturm-Liouville Problems, Differential and Integral Equations 1 (4) (1988), 423-432 · Zbl 0723.34023 [11] Marini, J. Diff. Equations 28 pp 1– (1978) [12] : Métodos da Teoria de Pontos Críticos, Textos de Matemática, Universidade de Lisboa, Faculdade de Ciências, Departamento de Matemática, 1993 [13] : Topological Vector Spaces, Springer-Verlag, New York, 1971. [14] : Spectral Theory of Ordinary Differential Operators, Lectures Notes in Mathematics 1258, Springer-Verlag, New York, 1987 [15] : Analyse Convexe et Optimisation, CIACO, Louvain-la-Neuve, 1989 [16] : Analyse Harmonique Réelle, Hermann, Paris, 1995 [17] : Sturm-Liouville Problems. In: Spectral Theory and Computational Methods of Sturm-Liouville Problems, edited by D. Hinton and P. W. Schaefer, pp. 1-104, Marcel Dekker, New York, 1997
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