Nilpotent ordinary differential operators with polynomial coefficients
We prove that nilpotent elements in the Weyl algebra have unique normal operator closures, and give a spectral representation. We consider the Schwartz space \({\mathcal S}\) as a dense subspace of \(L^ 2(R)\), and the operators \((Pf)(x)=-if'(x)\), \((Qf)(x)=xf(x)\), satisfying (ad P)(Q)f\(=[P,Q]f=PQf-QPf=-if\), \(f\in {\mathcal S}\), in the usual Schrödinger representation. Let W note the Weyl algebra generated by P and Q. We assume that some \(A\in W\) satisfies the local nilpotency condition: (LN): There is an integer n such that \((ad A)^ n(P)=(ad A)^ n(Q)=0.\) We then prove that there is a unitary operator U on \(L^ 2(R)\), and a complex polynomial in one variable \(\phi\), such that: (i) U(\({\mathcal S})={\mathcal S}\); (ii) \(\alpha =Ad(U)\) implements an automorphism of W; (iii) \(A=U\circ \phi (P)\circ U^*=\alpha (\phi (P))\) holds as an operator identity on \({\mathcal S}\). We also show that every \(\alpha\in Aut(W)\) satisfying \(\alpha (B^*)=\alpha (B)^*\), \(B\in W\), is unitarily implemented as in (i) and (ii). The result was conjectured by R. T. Powers. We apply this to give a new integrability theorem for the Heisenberg commutation relation, \([P,Q]=-i\).
scientific article; zbMATH DE number 3979532
scientific article