Indefinite Sturm-Liouville operators with the singular critical point zero

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.Comment: 24 pages, LaTeX2e <2003/12/01

From Sturm-Liouville problems to fractional and anomalous diffusions

Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the explicit laws of certain stable processes turn out to be fundamental. This paper directs one's efforts towards the explicit representation of solutions to fractional and anomalous diffusions related to Sturm-Liouville problems of fractional order associated to fractional power function spaces. Furthermore, we study a new version of the Bochner's subordination rule and we establish some connections between subordination and space-fractional operatorComment: Accepted by Stochastic Processess and Their Application