Spectral distributions and some Cauchy problems
A representation of the solution of the Cauchy problem when the operator admits a spectral distribution is given in this paper. The main theorem states: If \(B\) is the momentum of a spectral distribution \({\mathcal E}\) of degree \(k\), then for every \(x\in D(B^{k+1})\), \(s\in R\), \(u(s)= {\mathcal E}(t\mapsto (\lambda- t)^{- k-1}e^{ist})(\lambda- B)^{k+ 1}x\) is the unique solution of the Cauchy problem \[ {d\over dt} u(t)= iBu(t),\quad u(0)= x,\tag{1} \] and this solution is polynomially bounded, i.e., \(\|u(s)\|\leq C(1+|s|^k)\|(\lambda- B)^{k+1}x\|\) for every \(x\in\mathbb{R}\). Later the application to the Schrödinger operator on \(\mathbb{R}^N\) with and without potential, and then on a bounded domain, is given. The expression of the associated spectral distribution is given, too. A second application for the Dirac operator is given.
scientific article; zbMATH DE number 1369374
scientific article