The similarity problem for indefinite Sturm-Liouville operators with periodic coefficients (Q2890636)

Consider the differential expression \(l[y] := (-y'' + q(x) y)/\omega (x)\) on an interval \(I\) with real coefficients \(q, \omega \in L^1_{\mathrm{loc}}(I)\) satisfying \(\omega(x) \neq 0\) a.e. If \(\omega > 0\), then it is a classical result that \(l\) induces a selfadjoint operator \(L\) in the Hilbert space \(\mathcal K := L^2(I, \omega)\) with the inner product \([f,g] := \int f \overline{g} \; \omega\, dx\) (eventually adding boundary conditions). However, if \(\omega\) changes its sign, then \((\mathcal K, \; [\cdot, \cdot])\) is only a Krein space and \(L\) is definitizable in \((\mathcal K, \; [\cdot, \cdot])\). During the last decade, the question was intensively discussed whether \(L\) is at least \textit{similar} to a selfadjoint operator in a Hilbert space. Now, assume that \(L\) is J-positive, i.e., \([Lf,f] > 0\) for \(f \in \text{dom} (L) \setminus \{0\}\). Then, this so-called \textit{similarity property} is equivalent to the regularity of the critical point \(\infty\) and additionally, of \(0\) if \(0\) is a critical point of \(L\) at all. This is not the case if \(l\) is a regular differential expression, i.e., \(I\) is finite. The present paper studies the case when \(I\) is the whole real axis and \(q, \omega\) are \(2 \pi\)-periodic. It turns out that once \(0\) is a critical point, then it is a singular critical point and a more explicit characterization of this property is presented. This result gives rise to a new class of ``counterexamples'' for the similarity property, e.g., \(L = -(\operatorname{sgn} \sin x) \; d^2/dx^2\) on \(\mathbb R\). Furthermore, some known criteria for the regularity of the critical point \(\infty\) are generalized from the case of a regular differential expression \(l\) to the present \(2 \pi\)-periodic case. These conditions involve the behaviour of \(\omega\) near the turning points, i.e., the sign changes of \(\omega\) (assuming that there are only finitely many in \([0,2 \pi)\)).