Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution (Q1930793)

The aim of the paper is to find conditions assuring the property of basis for the system of eigenfunctions of the differential operator \[ Lu= -u''(-x),\quad -1< x< 1, \] in the complex space \(L_2(-1,+1)\), under the boundary value conditions (BVC) \[ \alpha_j u'(-1)+ \beta_j u'(1)+ \alpha_{j1} u(-1)+ \beta_{j1} u(1)= 0,\quad j= 1,2. \] A definition is given for regular and irregular type of BVC and a first result provides condtions for the Riesz basis property of the system of the eigenfunctions in the case of regularity of the BVC (not in the strong sense). In the irregular case, the system of eigenvalues is complete in the space \(L_2(-1,+1)\), but it does not form a basis (not necessarily a Riesz basis). Another result assures that the Riesz basis property is valid only in the case of regular conditions. Some interesting asymptotic formulas are provided for the eigenvalues.