Essential self-adjointness of operators associated with the Cauchy problem for the wave equation (Q1206307)

Given a function \(q\in C^\infty (\mathbb{R})\), let \(u(x, t)=: (U(t) f) (x)\) be a solution of the problem \(u_{tt}- u_{xx}+ qu =0\), \(u(x, 0)= f(x)\), \(u_t (x, 0) =0\), \(x\in \mathbb{R}\), \(t>0\). A sufficient condition is given that the operator \(U(t)\) is essentially selfadjoint for each \(t\in (0, l]\), where \(l>0\) is fixed.