Indefinite Sturm-Liouville operators with the singular critical point zero

Abstract

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 $L^2(\mathbb{R},(3|x|+1{)}^{-4/3}dx)$ 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.

24 pages, LaTeX2e <2003/12/01>

Cited by in corpus (5)

• The similarity problem for $J$-nonnegative Sturm-Liouville operators
• The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality
• Abstract kinetic equations with positive collision operators
• On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems
• A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators