Indefinite Sturm-Liouville operators with the singular critical point zero

Details Indefinite Sturm-Liouville operators with the singular critical point zero Indefinite Sturm-Liouville operators with the singular critical point zero

2006-12-06

arxiv.org/abs/math/0612173v1

Proceedings of the Royal Society of Edinburgh: Section A 138 (2008), no. 4 , pp. 801 -820

Mathematics

Spectral Theory

24 pages, LaTeX2e <2003/12/01>

Scientific paper

10.1017/S0308210507000157

We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh $m$-functions. Also we obtain necessary conditions for regularity of the critical points 0 and $\infty$ of $J$-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular critical point zero. In particular, it is shown that 0 is a singular critical point of the operator $-\frac{(\sgn x)}{(3|x|+1)^{-4/3}} \frac{d^2}{dx^2}$ acting in the Hilbert space $L^2(\R, (3|x|+1)^{-4/3}dx)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouville operator of type $(\sgn x)(-d^2/dx^2+q(x))$ with the same properties.

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