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

Stochastic Navier-Stokes equation and advection of a tracer field: One-loop renormalization near d=4d=4

The renormalization group approach and the operator product expansion technique are applied to the model of a tracer field advected by the Navier-Stokes velocity ensemble for a compressible fluid. The model is considered in the vicinity of the specific space dimension $d=4$. The properties of the equal-time structure functions are investigated. The multifractal behaviour of various correlation functions is established. All calculations are performed in the leading one-loop approximation.Comment: 8 pages, 1 figur