On double commutator relation (Q2711782)

The author considers representations of the relation \([a,[a,b]]=0\) by unbounded operators \(A,B\). It is assumed that \(A\) is self-adjoint and \(B\) is a closed extension of a densely defined symmetric operator. The commutation relation is understood as the pair of relations NEWLINE\[NEWLINE [E_A(\Delta) ,U_t^*E_A(\Delta')U_t]=0,\quad [E_A(\Delta),U_tE_A(\Delta')U_t^*]=0NEWLINE\]NEWLINE for any \(t\geq 0\) and any Borel sets \(\Delta ,\Delta'\). Here \(E_A\) is the spectral measure of \(A\), \(U_t=e^{itB}\). Some classes and specific examples of irreducible representations are given.