Sturm-Liouville operators with complex singular coefficients

Abstract

We consider on the finite interval the Sturm-Liouville differential expression
\[l(y)=-(py')'+qy+i((ry)'+ry')\]
with coefficients satisfying conditions: $q = Q',$ $1\Big/\sqrt{|p|},$ $Q\Big/\sqrt{|p|},$ $r\Big/\sqrt{|p|} \in L_2,$ where the derivative of function $Q$ is understood in the sense of distributions. Corresponding operators are correctly defined as quasi-differential. Conditions for the minimal operator to be symmetric are obtained and all its self-adjoint, maximal dissipative and maximal accumulative extensions are described in terms of boundary conditions.