Eigenfunction expansions for some nonselfadjoint operators and the transport equation (Q1059829)

Let A be an absolutely continuous selfadjoint operator acting in a Hilbert space H and let V be a closed linear operator which is A-bounded with A-bound less than 1. An eigenfunction expansion theorem is proved for the operator \(B=A+V\). The abstract results obtained are applied to the Schrödinger operator \(B=-\Delta +q\) in \(L^ 2({\mathbb{R}}^ 3)\) and also to the linear stationary transport equation \((\partial /\partial x)\mu \psi (x,\mu)=-(I+K)\psi +f,\) where I is the identity on \(H=L^ 2(- 1,1),\) K is a finite rank operator on H with \(\| K\| <1\), -1\(\leq \mu \leq 1\), \(x\geq 0\) and \(\psi (0,\mu)=g(\mu)\) and f(x,\(\mu)\) are known.