The similarity problem for $J$-nonnegative Sturm-Liouville operators

TL;DR

This paper establishes conditions under which certain indefinite Sturm-Liouville operators are similar to self-adjoint operators, focusing on the regularity of critical points and the role of the potential and weight functions.

Contribution

It provides new criteria based on Titchmarsh-Weyl m-coefficients for the similarity of J-nonnegative Sturm-Liouville operators to self-adjoint operators, including sharp conditions on the potential.

Findings

Operators with specific weight functions are similar to self-adjoint operators under certain conditions.

Regularity of critical points is characterized for various classes of Sturm-Liouville operators.

For periodic potentials, J-positivity ensures similarity to self-adjoint operators.

Abstract

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.