Spectral Properties of Differential–Difference Symmetrized Operators

We investigate some spectral properties of differential–difference operators, which are symmetrizations of differential operators of the form (𝔡^†𝔡)^k and (𝔡𝔡^† )^k , k⩾ 1 . Here, 𝔡=pd/dx +q and 𝔡^† stands for the formal adjoint of 𝔡 on L^2((0,b),w dx) . In the simpliest case k=1 , this symmetrization brings in the operator -𝔇^2 , which can be seen as a ‘Laplacian’, and 𝔇f:=𝔇_𝔡f= 𝔡(f_even)-𝔡^† (f_odd) , a skew-symmetric operator in L^2(I,w dx) , I=(- b,0)∪ (0,b) , is the symmetrization of 𝔡 . Investigated spectral properties include self-adjoint extensions, among them the Friedrichs extensions, of the symmetrized operators.