Maximal Accretive Realizations of Regular Sturm-Liouville Differential Operators

Abstract

Maximal accretive realizations and bound-preserving self-adjoint extensions are two fundamental problems in applications of semi-bounded operator theory to differential equations. On the basis of using differential operator theory in direct sum spaces and Phillips theory for maximal accretive extensions of accretive operators, a complete characterization of the set of maximal accretive boundary conditions for Sturm–Liouville differential operators is presented. As an application, all possible forms of bound-preserving self-adjoint extensions of regular Sturm–Liouville operators are also characterized via various explicit boundary conditions. The methodology can also be applied to dealing with general classes of semi-bounded symmetric differential operators.