On the unboundedness below of the Sturm-Liouville operator. (English) Zbl 0947.34010
It is very natural to suppose that the semiboundedness of the minimal operator \(T_0\) associated with \(w^{-1} [-(py')' +qy]\) in \(L^2(I;w)\), \(I\) an arbitrary interval, will be destroyed by a leading coefficient \(p\) that is negative on a set of positive Lebesgue measure. (The local regularity conditions on the coefficients are the minimal ones, but the functions \(q\) and \(w\) do not really matter.)
The author proves this by using a function \(0\leq \varphi \in C^\infty_0 ({\overset \circ I})\) such that \(\int_I \varphi/p<0\), the technical difficulty being that \(\varphi\) is in general not in the domain of \(T_0\).
The author proves this by using a function \(0\leq \varphi \in C^\infty_0 ({\overset \circ I})\) such that \(\int_I \varphi/p<0\), the technical difficulty being that \(\varphi\) is in general not in the domain of \(T_0\).
Reviewer: Hubert Kalf (München)
MSC:
| 34B24 | Sturm-Liouville theory |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
References:
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| [5] | Daho, Proc. R. Soc. Edinb. A 78 pp 161– (1977) · Zbl 0372.34018 · doi:10.1017/S0308210500009914 |
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