Sufficient conditions for the self-adjointness of the Sturm—Liouville operator

Let L be the minimal operator in L₂(R¹) generated by the expressionly=?y″+q(x)y, Im q(x) ≡ 0, let Δ_k(k=+-1,+-2,...) be a sequence of disjoint intervals going out to +-∞ for k→+=∞, and let δ_k be the length Δ_k. If (ly,y)^≥?γ_k‖y‖² on all smooth y(x) with support in δ_k, wherebyγ _k>0, $$sumnolimits_{k = 1}^infty {(gamma _k + delta _k^{ - 2} ) - 1 = } sumnolimits_{k = - infty }^{ - 1} {(gamma _k + delta _k^{ - 2} ) - 1 = infty ,} $$ . then the operator L is self-adjoint. This theorem generalizes criteria for the self-adjointness of L obtained earlier by R. S. Ismagilov, A. Ya. Povzner, and D. B. Sears.