Indefinite Sturm-Liouville operators with periodic coefficients (Q2873566)

The maximal operator \(A\) associated with the differential expression NEWLINE\[NEWLINE\mathfrak t (f) := (-(pf')' + q f)/wNEWLINE\]NEWLINE on the whole real line and with real coefficients \(w, p, q\) is so far well understood in the right definite case (i.e. \(w > 0\)) and in the left definite case (i.e. \(\int {\mathfrak t} (f) \overline{f} \, w \; dx \geq 0\)). In the present paper these definiteness conditions are dropped and and it is only assumed that \(p > 0, w \neq 0\) a.e. and \(w, 1/p, q\) are locally integrable and periodic with a common period \(a > 0\). Then, due to the limit point case, the operator \(A\) is J-self-adjoint in the weighted Krein space \(L^2_w({\mathbb R})\) with the inner product \([f,g] := \int f \overline{g} \, w \; dx\). Although, in general \(A\) is not definitizable its spectral properties have certain analogies to definitizable operators.NEWLINENEWLINE It is shown that the real line can be devided into finitely many subintervals which are either of positive or negative spectral type and it is possible to introduce a spectral function \(E(\Delta)\) with critical points which are either regular or singular (where \(||E(\Delta)||\) is unbounded). Some necessary and sufficient conditions for the regularity of finite critical points are given. Furthermore, for the point \(\infty\) a sufficient regularity condition is obtained, generalizing some results from the definitizable case.NEWLINENEWLINE However, the main difference to the definitizable case is the appearance of a bounded non real spectrum consisting of symmetric analytic curves. By the Gelfand transform this result is reduced to the definitizable operator \(A(t)\) in the Krein space \(L^2_w(0,a)\) given by \({\mathfrak t} (f)\) on \((0,a)\) and the boundary conditions NEWLINE\[NEWLINEf(a) = e^{it}f(0),\quad (pf')(a) = e^{it}(pf')(0).NEWLINE\]NEWLINE Indeed, it turns out that \(\sigma (A) = \bigcup \sigma (A(t))\) where \(t\) runs through \([-\pi, \pi]\). The question remains open whether in general the class of definitizable operators in Krein spaces can be generalized to a class of operators with the above properties such that \(A\) can serve as an example.