Spectral representation of a hyperbolic evolutionary process connected with a Sturm-Liouville differential expression (Q753015)

The author investigates sufficient conditions which guarantee existence of a spectral representation \(U(t)=\cos (tA^{1/2})\), \(t>0\), for the Cauchy problem (for a hyperbolic equation): \[ \partial^ 2u/\partial t^ 2+S[u]=0,\quad u|_{t=0}=f\in C^{\infty}_ 0({\mathbb{R}}),\quad \partial u/\partial t|_{t=0}=0, \] where \(S=-d^ 2/dx^ 2+q(x)\) is the Sturm-Liouville differential expression.