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. We also obtain necessary conditions for regularity of the critical points $0$ and $\infty$ of $J$-non-negative Sturm–Liouville operators. Using this result, we construct several examples of operators with the singular critical point $0$. In particular, it is shown that $0$ is a singular critical point of the operator

$$ -\frac{(\mathrm{sgn} x)}{(3|x|+1)^{-4/3}}\frac{\mathrm{d}^2}{\mathrm{d} x^2} $$

acting in the Hilbert space $L^2(\mathbb{R},(3|x|+1)^{-4/3}\,\mathrm{d} x)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a $J$-non-negative Sturm–Liouville operator of type $(\mathrm{sgn} x)(-\mathrm{d}^2/\mathrm{d} x^2+q(x))$ with the same properties.