A new characterization of the Friedrichs extension of semibounded Sturm- Liouville operators. (English) Zbl 0615.34019

Let \({\mathcal A}\) denote the Sturm-Liouville differential expression given by \({\mathcal A}u:=(1/k)\{-(pu')'+qu\}\) on \(I:=(\ell,m)\), where \(-\infty \leq \ell <m\leq \infty\) and the coefficients of \({\mathcal A}\) satisfy the conditions: (i) k,1/p,q, \(L^ 1_{loc}(I)\) and are real-valued, (ii) k, \(p>0\) a.e. on I. The minimal operator A associated with \({\mathcal A}\) in \({\mathcal H}=L^ 2(I,k)\) is defined by \(Au:={\mathcal A}u\), \(D(A):=\{u| u\in {\mathcal D}\), supp \(u\subset I\) compact\(\}\), where \({\mathcal D}:=\{u| u\in {\mathcal H}\), \(u,pu'\in AC_{loc}(I)\), \({\mathcal A}u\in {\mathcal H}\}\). If A is bounded below it has a Friedrichs extension \(A_ F\) and for any \(u\in D(A_ F)\) there exists a sequence \(\{u_ j\}\) in D(A) with \(u_ j\to u\) in \({\mathcal H}\) and \((A_ Fu,u)=\lim_{j\to \infty}\int^{m}_{\ell}(p| u_ j'|^ 2+q| u_ j|^ 2).\) The paper examines the existence of \(\int^{m}_{p}| u'|^ 2+q| u|^ 2\) (defined as an improper integral) for all \(u\in D(A_ F):\) it is well-known from examples of Rellich and Moser that this is not true in general. Assuming (for simplicity) that m is a regular point, conditions are obtained which ensure that \[ D(A_ F)=M_ 0:=\{u| u\in {\mathcal D},\quad u(m)=0,\quad \lim_{x\to \ell +}\int^{m}_{x}(p| u'|^ 2+q| u|^ 2)\quad exists\}: \] the assumption that m is regular imposes the condition \(u(m)=0\) on any \(u\in D(A_ F)\). It is proved that, if \({\mathcal A}\) is in the limit circle (LC) case at \(\ell\), then \(D(A_ F)=M0\) if and only if \((i)\quad \lim_{x\to \leftrightarrow +}pff'(x)\) exists and \((ii)\quad \lim_{x\to \leftrightarrow +}pgg'(x)\) does not exist, where f is a principal solution and g a non-principal solution of (\({\mathcal A}- \lambda)u=0\) at \(\ell\) for some \(\lambda\in {\mathbb{R}}\). If Hinton’s condition \(pff'(x)=O(f(x)/g(x))\) as \(x\to \ell +\) is satisfies and either \(\lim_{x\to \leftrightarrow +}pgg'(x)\) does not exist or \({\mathcal A}\) is in the limit (LP) case at \(\ell\), then \[ D(A_ F)=\tau_ 0:=\{u| u\in {\mathcal D},\quad u(m)=0,\quad \int^{m}_{\ell}p| u'|^ 2<\infty,\quad \lim_{x\to \ell +}\int^{m}_{x}q| u|^ 2\quad exists\}. \] This result ceases to hold if \({\mathcal A}\) is LP at \(\ell\) and Hinton’s condition is weakened to lim pff’(x)\(=0\); indeed \(D(A_ F)\subset M_ 0\) is not guaranteed in these circumstances at the author proves. Another interesting result is that \(D(A_ F)=M_ 0\) if \(\ell\) is not a regular point and is a singular point of the first kind (a regular singular point) for (\({\mathcal A}-\lambda)u=0\) (\(\lambda\in {\mathbb{C}})\). It is shown that this assumption does not imply the existence of the separate integrals \(\int^{m}_{\ell}p| u'|^ 2\), \(\int^{m}_{\ell}q| u|^ 2\) for all \(u\in D(A_ F)\). The implication is true if \({\mathcal A}\) is LP at \(\ell\); in fact \(\int^{m}_{\ell}| q| | u|^ 2<\infty\) in this case. When A is LC at \(\ell\), \(D(A_ F)=\{u| u\in {\mathcal D}\), \(u(m)=0\), \(\int^{m}_{\ell}| p| u'|^ 2+q| u|^ 2| <\infty \}\).
Reviewer: W.D.Evans


34L99 Ordinary differential operators
47B44 Linear accretive operators, dissipative operators, etc.
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