Singular perturbations in the interaction representation, II
This paper is a sequel and generalization of a previous paper [J. Funct. Anal. 38, 71-98 (1980; Zbl 0454.47014)] by the same author and \textit{I. E. Segal} and is devoted to the definition and construction of the unitary one-parameter group U(t) in \(L^ 2({\mathbb{R}})\) (more generally \(L^ 2({\mathbb{R}}^ k))\) formally generated by the operator \(H=H_ 0+V\) where \(H_ 0=(-id/dx)^ m\) in \(L^ 2({\mathbb{R}})\) (more generally \(H_ 0=(- \Delta)^{m/2}\), \(m\geq 2\), in \(L^ 2({\mathbb{R}}^ k))\), and V is a real distribution in \({\mathcal S}'({\mathbb{R}})\) (or \({\mathcal S}'({\mathbb{R}}^ k))\) with Fourier transform \(\hat V\in L^{\infty}_{loc}\) and prescribed polynomial growth at infinity in moment space. The method is the same as in the previous paper and consists in proving the norm convergence for small time intervals of the perturbation series for the two parameter unitary group \[ U(t;s)=\exp(it H_ 0)\quad \exp(-i(t-s)H)\exp(is H_ 0), \] generated by H in the so called interaction representation familiar in Quantum Mechanics (see the abstract of the first paper for more detail). In dimension 1, the result holds under the growth condition on \^V: \(| \hat V(\xi)| \leq M(1+| \xi |^{\alpha})\quad with\quad \alpha =(m-(1+\epsilon))/2,\quad 0<\epsilon \leq m-1.\) In addition, U is continuous as a function of V in the uniform operator topology for \(\hat V\) converging uniformly in \({\mathbb{R}}\), and in the strong operator topology for \(\hat V\) converging pointwise almost everywhere and satisfying the previous estimate with fixed M and \(\alpha\). The analogous results in dimension \(k>1\) are briefly mentioned in the last section of the paper. The proof consists in momentum space estimates of the integral kernels of the operator \[ \int^{t}_{0}ds \exp(is H_ 0)V \exp(-is H_ 0) \] and suitable iterates thereof, which constitute the terms in the perturbation series for U(t,s).
scientific article; zbMATH DE number 3842537