## 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

### MSC:

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