The similarity problem for JJ-nonnegative Sturm-Liouville operators

Sufficient conditions for the similarity of the operator $A := 1/r(x) (-d^2/dx^2 +q(x))$ with an indefinite weight r(x)=(\sgn x)|r(x)| are obtained. These conditions are formulated in terms of Titchmarsh-Weyl $m$-coefficients. Sufficient conditions for the regularity of the critical points 0 and $\infty$ of $J$-nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r(x)=\sgn x and $q\in L^1(R, (1+|x|)dx)$, we prove that $A$ is similar to a self-adjoint operator if and only if $A$ is $J$-nonnegative. The latter condition on $q$ is sharp, i.e., we construct $q\in \cap_{\gamma <1} L^1(R, (1+|x|)^\gamma dx)$ such that $A$ is $J$-nonnegative with the singular critical point 0. Hence $A$ is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that $J$-positivity is sufficient for the similarity of $A$ to a self-adjoint operator. In the case $q\equiv 0$, we prove the regularity of the critical point 0 for a wide class of weights $r$. This yields new results for "forward-backward" diffusion equations.Comment: 36 pages, LaTeX2e, version 2; addresses of the authors added, the reference [38] update

On an oblique derivative problem involving an indefinite weight

summary:In this paper we derive results concerning the angular distrubition of the eigenvalues and the completeness of the principal vectors in certain function spaces for an oblique derivative problem involving an indefinite weight function for a second order elliptic operator defined in a bounded region