Indefinite Sturm-Liouville operators in polar form

Abstract

Let B + and B − be nonnegative symmetric operators with defect numbers (1, 1) in Hilbert spaces H + and H − . We consider the coupling A of operators A + = B + and A − = −B − which is self-adjoint, nonnegative and has a nonempty resolvent set in the Kreȋn space K with the fundamental decomposition K = H + [+]H − . Such an operator is definitizable and has at most two critical points at ∞ and 0. It is similar to a self-adjoint operator in a Hilbert space if and only if its critical points are regular and there are no Jordan chains at 0. We find criteria for the regularity of critical points of A formulated in terms of the abstract Weyl functions m + and m − of the operators B + and B − .A typical example of such coupling is the operator A generated in the Kreȋn space K = L 2 w (I) by the differential expression> 0 and (sgn t)w(t) > 0 a.e. on I. The operator A is self-adjoint in the Kreȋn space L 2 w (I) and can be considered as the coupling of the restrictions of A onto L 2 w − (I − ) and L 2 w + (I + ), where I − = (b − , 0), I + = (0, b + ), w − is the restriction of −w onto I − and w + is the restriction of w onto I + . For this operator the general criteria for the regularity of a critical point are reformulated in terms of the coefficients w and r.