Spectral Properties of Differential–Difference Symmetrized Operators

Abstract

We investigate some spectral properties of differential–difference operators, which are symmetrizations of differential operators of the form (mathfrak {d}^dagger mathfrak {d})^k and (mathfrak {d}mathfrak {d}^dagger )^k, kgeqslant 1. Here, mathfrak {d}=pfrac{d}{textrm{d}x} +q and mathfrak {d}^dagger stands for the formal adjoint of mathfrak {d} on L^2((0,b),w,textrm{d}x). In the simpliest case k=1, this symmetrization brings in the operator -mathfrak {D}^2, which can be seen as a ‘Laplacian’, and mathfrak {D}f:=mathfrak {D}_{mathfrak {d}}f= mathfrak {d}(f_{text {even}})-mathfrak {d}^dagger (f_{text {odd}}), a skew-symmetric operator in L^2(I,w,textrm{d}x), I=(-,b,0)cup (0,b), is the symmetrization of mathfrak {d}. Investigated spectral properties include self-adjoint extensions, among them the Friedrichs extensions, of the symmetrized operators.