Strongly singular Sturm-Liouville problems (Q2732531)

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 NEWLINE\[NEWLINE-(ru'(x))'+ p(x) u(x)=\lambda m(x) u(x)\quad\text{on }(a,b)NEWLINE\]NEWLINE 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.NEWLINENEWLINENEWLINEThus, 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.NEWLINENEWLINENEWLINEHere, \(r: (a,b)\to \mathbb{R}^+\) 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.NEWLINENEWLINENEWLINEThe 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 NEWLINE\[NEWLINE\cdots<\lambda_{-2}< \lambda_{-1}< 0< \lambda_1< \lambda_2<\cdots\;.NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE\min\left\{ \int^b_a (ru^{\prime 2}+ pu^2)\Biggl/ \int^b_a mu^2;\;\begin{matrix} u\in C[a,b),\;u(a)= 0\\ \sqrt pu,\sqrt ru'\in L^2(a,b), \int^b_a mu^2> 0\end{matrix}\right\}.NEWLINE\]NEWLINE 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.