Spectral Properties of Differential–Difference Symmetrized Operators

Abstract

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