Scale-invariant self-adjoint extensions of scale-invariant symmetric operators. (English) Zbl 1340.47046

A closed symmetric operator \(A\) on a Hilbert space \(\mathcal H\) with the domain \(D(A)\) is called \((q^2,U)\)-scale invariant, if \(q>0\), \(q\neq 1\), \(U\) is a unitary operator, \(UD(A)=D(A)\), and \(UAf=q^2AUf\) for each \(f\in D(A)\). The authors prove that there are exactly four possibilities: the operator \(A\) has no \((q^2,U)\)-scale invariant selfadjoint extensions, or exactly one, two, or all its selfadjoint extensions possess this property. Some classes of differential and \(q\)-difference operators are considered, for which the above properties hold.


47B25 Linear symmetric and selfadjoint operators (unbounded)
47E05 General theory of ordinary differential operators
39A13 Difference equations, scaling (\(q\)-differences)
47B39 Linear difference operators