48 research outputs found
The similarity problem for -nonnegative Sturm-Liouville operators
Sufficient conditions for the similarity of the operator with an indefinite weight r(x)=(\sgn x)|r(x)| are
obtained. These conditions are formulated in terms of Titchmarsh-Weyl
-coefficients. Sufficient conditions for the regularity of the critical
points 0 and of -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 , we prove that is similar to a
self-adjoint operator if and only if is -nonnegative. The latter
condition on is sharp, i.e., we construct such that is -nonnegative with the singular critical
point 0. Hence is not similar to a self-adjoint operator. For periodic and
infinite-zone potentials, we show that -positivity is sufficient for the
similarity of to a self-adjoint operator. In the case , we prove
the regularity of the critical point 0 for a wide class of weights . 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